The last of the high-frequency Algebra types (4.2% of the bank): inequalities in one or two variables. Covers solving exactly as you would an equation with the one exception that multiplying or dividing by a negative flips the sign, translating at most and at least into the right symbols, representing solutions on a number line, and identifying shaded half-planes and systems of them by testing the origin — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 13 of 19
4.2% of the question bank — 71 of 1675 questions
Objectives
An inequality is an equation with a direction attached. Solving one uses exactly the same moves, with a single exception that reverses the direction. The second half of this type is different in character: a shaded region on a graph, where the question is which side of a line satisfies the condition, and the answer comes from testing one point.
Two rules carry the entire type: flip when you multiply or divide by a negative, and test the origin to find the shaded side.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 71 tagged questions of this type in the site's bank
Section
Section 1
Concept
An inequality is an equation with a direction attached. Solving one uses exactly the same moves, with a single exception that reverses the direction. The second half of this type is different in character: a shaded region on a graph, where the question is which side of a line satisfies the condition, and the answer comes from testing one point.
You will see it phrased in these ways:
The region questions are roughly a third of this type and are the part students practise least. They are also the fastest, because testing (0, 0) settles nearly all of them.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Linear inequalities in one or two variables: the tell, the move, and the trap
The green panel contains the two whole methods of this type. Everything else in the deck is the detail of applying them.
Prediction
One rule, and it applies less often than students fear.
Predict first
Which operation requires you to reverse the inequality sign?
Correct: Dividing both sides by negative 3
Why: Only multiplying or dividing by a NEGATIVE number reverses the direction. Adding and subtracting never do, whatever the sign of what you are adding, and multiplying or dividing by a positive never does either. The reason is that multiplying by a negative reflects the number line: 2 is less than 5, but negative 2 is greater than negative 5.
Concept
Three shapes of question, and the last two are quite different in method.
| variant | what it wants | the move |
|---|---|---|
| Solve in one variable | a range of values for x | solve like an equation, watching for the flip |
| A word problem | a maximum or minimum quantity | translate the words into a symbol, then solve |
| A region in the plane | which inequality matches a shaded graph | test the point (0, 0) |
The third variant looks the hardest and is the fastest. One substitution decides which side of a line is shaded, and a second decides a system.
Definition probe
Translation is where word problems on this type are won or lost.
Sort into buckets
Which inequality does each phrase become?
Discrimination
Six final steps. Which reverse the sign?
Sort into buckets
Does the inequality sign flip?
Warm-up
Try it, and watch the last step.
Discussion prompt
Solve negative 3x plus 5 is greater than 17.
Hint: Which step involves a negative multiplier?
Answer:
Subtract 5 from both sides: negative 3x is greater than 12. No flip — subtraction never flips.
Divide both sides by negative 3: this is the step that flips. So x is less than negative 4.
Check it with a value: try x equals negative 5, which satisfies x less than negative 4. Then negative 3 times negative 5 plus 5 is 15 plus 5, which is 20, and 20 is greater than 17. Correct.
Check the other side: try x equals 0. Then 5 is not greater than 17, so 0 correctly fails.
Substituting one value from each side of the boundary is the reliable way to confirm the direction, and it takes about ten seconds.
Pattern
Three steps for a one-variable solve, and a different one-step method for regions.
Solve exactly as if the inequality sign were an equals sign.
Why: Every algebraic move is identical: clear fractions, expand brackets, gather the variable. Nothing changes until the final division.
At each multiplication or division, check the sign of what you are multiplying by, and flip if it is negative.
Why: This is the only difference from an equation, and it applies at the moment of the operation rather than at the end.
Test one value from each side of your boundary in the ORIGINAL inequality.
Why: One value that should work and one that should not confirms both the boundary and the direction, which is exactly what can go wrong.
For a shaded region the routine is shorter still: substitute (0, 0) and see whether the inequality holds. If it does, the origin's side is shaded.
Section
Section 2
Concept
Every step you would use on an equation is legal on an inequality.
Because the moves are identical, everything you already know from type 7 transfers directly. Only the final division needs new attention.
Concept
The one exception: a negative multiplier reverses the direction of the inequality.
| operation | example | flips? |
|---|---|---|
| add or subtract anything | x minus 3 is less than 5 | no |
| multiply or divide by a positive | 2x is less than 8 | no |
| multiply or divide by a negative | negative 2x is less than 8 | YES |
That last point is a genuine tactic: from negative 3x greater than 12, adding 3x to both sides and moving the 12 across avoids the flip entirely.
Prediction
Watch the divisor.
Predict first
Solving negative 2x is greater than 10 gives which result?
Correct: x is less than negative 5
Why: Dividing both sides by negative 2 flips the sign, giving x less than negative 5. Checking with x equals negative 6: negative 2 times negative 6 is 12, and 12 is greater than 10, so it works. Checking x equals 0: 0 is not greater than 10, so it correctly fails. Keeping the sign unflipped gives the second choice, which fails both tests.
Concept
At most and at least include the value; more than and fewer than exclude it.
| phrase | symbol | is the value itself allowed? |
|---|---|---|
| at most 20, no more than 20, a maximum of 20 | less than or equal to | yes |
| at least 20, no fewer than 20, a minimum of 20 | greater than or equal to | yes |
| more than 20, greater than 20 | greater than | no |
| fewer than 20, less than 20, under 20 | less than | no |
Read the phrase, decide whether the endpoint is allowed, and only then choose between the two symbols.
Prediction
Does the boundary count?
Predict first
A lift holds at most 8 people. Which inequality models the number n of people?
Correct: n is less than or equal to 8
Why: At most 8 means 8 is permitted and nothing above it, so the equal bar is included. Choosing strict less than would forbid exactly 8 people, which contradicts the ordinary meaning of a capacity. The two greater-than options reverse the direction entirely, describing a minimum rather than a maximum.
Concept
A filled circle includes the endpoint; an open circle excludes it.
The same convention appears on a graph: a solid boundary line includes the line, and a dashed one excludes it.
Concept
Substitute (0, 0) into the inequality. If it is true, the side containing the origin is shaded.
One substitution answers a question that looks geometric. This is the single most efficient move in the type.
Prediction
One substitution decides the side.
Predict first
For the inequality y is greater than 2x plus 1, is the point (0, 0) in the shaded region?
Correct: no
Why: Substituting gives 0 greater than 2 times 0 plus 1, which is 0 greater than 1 — false. So the origin is not in the shaded region, and the shading is on the other side of the line. The origin is not on the boundary either, since the line passes through (0, 1) rather than (0, 0).
Concept
A solid line means the boundary itself satisfies the inequality; a dashed line means it does not.
Read the line style first and the shading second. The line style halves the choices at no cost.
Concept
Two shaded regions together mean the points that satisfy both, which is where the shadings overlap.
For a which-point-is-a-solution question, substitution beats reading the graph. Check each choice against each inequality and eliminate on the first failure.
Prediction
Solid or dashed, before anything else.
Predict first
A graph shows a dashed boundary line with shading above it. Which symbol must the inequality use?
Correct: greater than
Why: A dashed line excludes the boundary, so the symbol must be strict — that rules out both options with an equal bar. Shading above the line means the y-values are larger than those on the line, so the symbol is greater than. Reading the line style first eliminates half the choices before you even consider the shading.
Two truths and a lie
Three of these statements about inequalities are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: Only multiplying or dividing by a NEGATIVE number flips the sign. Multiplying by a positive leaves the direction untouched: from x less than 3, multiplying by 2 gives 2x less than 6, which is still correct. The precision matters because students who over-apply the rule flip on every multiplication and reverse perfectly good answers.
Check
Watch the final division, then test a value.
Check your understanding
What is the solution to 5 minus 3x is less than or equal to 20?
Answer: A
Why: Subtracting 5 gives negative 3x less than or equal to 15. Dividing by negative 3 flips the sign, giving x greater than or equal to negative 5. Testing x equals 0: 5 minus 0 is 5, and 5 is less than or equal to 20, so 0 correctly satisfies it — and 0 is indeed greater than negative 5.
Testing x equals 0 separates A from B in about five seconds, and those are the only two choices a careful solver is deciding between.
Section
Section 3
Worked example
Solve 7 minus 2x is greater than or equal to 19.
Figure (svg): A number line shaded to the left of negative six with a filled circle at negative six
Subtract 7 from both sides: negative 2x is greater than or equal to 12.
Why: Subtraction never flips the sign.
Divide both sides by negative 2, which flips the sign: x is less than or equal to negative 6.
Why: Dividing by a negative reverses the direction of the inequality.
Represent it: filled circle at negative 6, shaded to the left.
Why: The equal bar makes the endpoint included, so the circle is filled.
Testing one value from each side is what confirms the direction. Getting the boundary right and the direction wrong is the commonest way to lose this question.
Verify: test x equals negative 7: 7 minus negative 14 is 21, and 21 is greater than or equal to 19.
Why: A value on the shaded side satisfies the original, and x equals 0 gives 7, which correctly fails.
Answer: x is less than or equal to negative 6
Worked example
Solve 12 minus 5x is less than 2, without ever dividing by a negative.
Figure (svg): A number line shaded to the right of two with an open circle at two
Add 5x to both sides: 12 is less than 2 plus 5x.
Why: Moving the variable term to the right keeps its coefficient positive, so no flip will be needed.
Subtract 2: 10 is less than 5x.
Why: Subtraction never flips.
Divide by positive 5: 2 is less than x, which is the same as x greater than 2.
Why: Dividing by a positive does not flip, so no reversal occurs anywhere.
This route never risks the flip error at all. When the variable has a negative coefficient, moving it across is often the safer choice.
Verify: test x equals 3: 12 minus 15 is negative 3, and negative 3 is less than 2.
Why: A value above 2 satisfies the original, and x equals 0 gives 12, which correctly fails.
Answer: x is greater than 2
Worked example
A delivery van weighs 1,200 kg empty and can carry a total loaded weight of at most 2,500 kg. Each crate weighs 85 kg. How many crates can it carry?
Figure (svg): A number line shaded from zero to fifteen with a filled circle at fifteen
Translate: at most 2,500 means the total is less than or equal to 2,500.
Why: At most includes the boundary, so the equal bar is part of the model.
Write it: 1200 plus 85c is less than or equal to 2500, so 85c is less than or equal to 1300.
Why: The empty weight is fixed and the crates add 85 kg each.
Divide by positive 85: c is less than or equal to about 15.29. Since crates are whole, the maximum is 15.
Why: Dividing by a positive does not flip, and the context requires an integer.
The rounding here goes DOWN even though the arithmetic gives 15.29, because 16 crates would exceed the limit. Context decides the rounding direction, not the decimal.
Verify: check 15 crates: 1200 plus 1275 is 2475, which is under 2500. Check 16: 1200 plus 1360 is 2560, over the limit.
Why: Fifteen fits and sixteen does not, confirming the maximum.
Answer: at most 15 crates
Step zero
On a shaded-region question, before anything else.
Discussion prompt
A graph shows a shaded half-plane and four inequalities as choices. What two things do you read off before doing any substitution?
Hint: One is about the line and one is about its style.
Answer:
First, the equation of the boundary line — its slope and its y-intercept. That narrows the choices to those with the right line.
Second, whether the line is solid or dashed. Solid means the symbol includes equality; dashed means it is strict.
Those two readings often eliminate three of the four choices before you substitute anything.
Then test (0, 0) to settle which side is shaded, unless the boundary passes through the origin — in which case pick any other convenient point.
The order matters because the first two steps are free readings and the third is the only one that requires work.
Worked example
A graph shows a solid line through (0, 3) with slope negative 1, and the region below it is shaded. What is the inequality?
Figure (svg): A solid line with the region below it shaded and the origin marked as a test point
The line has slope negative 1 and intercept 3, so its equation is y equals negative x plus 3.
Why: Slope-intercept form from the two visible facts.
The line is solid, so the inequality includes equality — the symbol has an equal bar.
Why: Solid boundaries are included; dashed ones are not.
Test (0, 0): is 0 less than or equal to 3? Yes, so the origin is in the region, and the shading is below.
Why: The origin satisfies the less-than-or-equal version, matching the shaded side.
Two facts were read off before any testing: the line's equation and its style. The origin test then decided the only remaining question.
Verify: test a point above the line, such as (0, 5): is 5 less than or equal to 3? No.
Why: A point on the unshaded side correctly fails, confirming the direction.
Answer: y is less than or equal to negative x plus 3
Worked example
Which of the points (0, 0), (4, 1) and (1, 4) satisfies both y greater than x minus 2 and y less than 3?
Figure (svg): Two boundary lines with the overlapping region between them
Test (0, 0): is 0 greater than negative 2? Yes. Is 0 less than 3? Yes. Both hold.
Why: A point is a solution only if it satisfies every inequality in the system.
Test (4, 1): is 1 greater than 2? No. It fails the first, so stop.
Why: One failure is enough to eliminate a candidate; there is no need to check the second condition.
Test (1, 4): is 4 greater than negative 1? Yes. Is 4 less than 3? No. It fails the second.
Why: Passing one condition is not sufficient.
Substitution beat reading the graph here, and eliminating on the first failure saved half the work.
Verify: check that (0, 0) sits in the overlap on the graph.
Why: The origin lies above the line y equals x minus 2 and below the line y equals 3, which is the overlapping wedge.
Answer: only (0, 0)
Faded example
From memory. Four lines that carry the type.
Fill in the blanks
The inequality sign flips only when you multiply or divide by a negative. At most means less than or equal to. On a number line, a filled circle means the endpoint is included. And to find which side of a line is shaded, test the point (0, 0).
Why: The first and last are the two methods of the type, and they belong to its two halves. Everything else is either translation from words or notation on a graph, both of which are conventions rather than mathematics.
Worked example
Solve negative 4 is less than 2x plus 6, which is less than or equal to 10.
Figure (svg): A number line shaded between negative five and two, open at the left and closed at the right
Subtract 6 from all three parts: negative 10 is less than 2x, which is less than or equal to 4.
Why: Whatever you do to one part of a compound inequality, do to all three.
Divide all three parts by positive 2: negative 5 is less than x, which is less than or equal to 2.
Why: Dividing by a positive does not flip any of the signs.
Represent it: open circle at negative 5, filled circle at 2, shaded between.
Why: The left symbol is strict and the right one includes equality, so the endpoints differ.
The two endpoints have different styles because the two symbols differ. Copying one style to both is an easy slip on compound inequalities.
Verify: test x equals 0: 2 times 0 plus 6 is 6, and negative 4 is less than 6, which is less than or equal to 10.
Why: A value inside the interval satisfies both parts, and x equals 3 gives 12, which correctly fails the upper bound.
Answer: negative 5 is less than x, which is less than or equal to 2
Fill the middle
The notation is a language; read it deliberately.
Fill in the blanks
A solid boundary line means the points on the line are included. A dashed boundary line means they are not. On a number line, an open circle corresponds to a strict inequality. And a system of two inequalities is solved by the region where the shadings overlap.
Why: Reading the line style first is worth building into the routine, because it decides between the two symbols before you have thought about direction at all, and on a four-choice question that usually halves the field for free.
Estimation
One substitution catches a missed flip.
Predict first
You solve negative 5x is greater than 20 and get x greater than negative 4. Testing x equals 0 gives what?
Correct: 0 is not greater than 20, so the answer is wrong
Why: The claimed solution x greater than negative 4 includes x equals 0, so 0 should satisfy the original inequality. Substituting gives negative 5 times 0, which is 0, and 0 is not greater than 20 — so it fails. That proves the direction was reversed: the flip was missed, and the correct answer is x less than negative 4. One substitution catches the error in about five seconds.
Check
Decide whether the boundary counts.
Check your understanding
A student needs an average of at least 85 across four tests. She has scored 82, 88 and 91. What score s does she need on the fourth?
Answer: A
Why: At least 85 means the average must be greater than or equal to 85, so the total must be at least 340. The three known scores sum to 261, so s must be at least 79. Checking: 261 plus 79 is 340, and 340 over 4 is exactly 85, which the phrase at least permits.
Choices A and B differ only in the equal bar, and the deciding word is at least. That single distinction is the whole question.
Section
Section 4
Trap
The trap. You solve negative 2x greater than 8 and write x greater than negative 4.
Dividing by negative 2 reverses the direction, so the answer is x LESS than negative 4.
The boundary is right and the direction is wrong, which means the answer looks entirely reasonable.
The fix. At the moment you divide, look at the sign of the divisor and flip if it is negative.
Then test one value from each side of the boundary in the original inequality.
Trap
The trap. Over-applying the rule, you flip whenever a negative appears anywhere — including when subtracting a number or adding a negative.
From x plus 5 less than 2, you subtract 5 and write x greater than negative 3.
Adding and subtracting never flip, whatever the signs involved. The answer is x less than negative 3.
The fix. The rule is about the OPERATION, not about whether a minus sign is visible.
Only multiplication and division by a negative flip the direction.
Error analysis
A student's solve. Every arithmetic step is right and the answer is wrong.
Annotate
On: \( -3x + 4 \;>\; 19 \;\Rightarrow\; -3x \;>\; 15 \;\Rightarrow\; x \;>\; -5 \)
On this type the boundary is usually right and the direction is what fails. That is precisely why the check is to substitute a value on each side rather than to re-do the algebra.
Trap
The trap. A question says a budget is at most 500 dollars and you write strictly less than 500.
At most includes the boundary, so spending exactly 500 is allowed and the symbol needs the equal bar.
The answer choices will offer both versions, differing in nothing else.
The fix. Ask whether hitting the limit exactly is permitted by the story.
At most, at least, no more than, minimum and maximum all include the boundary; more than and fewer than do not.
Elimination
A graph shows a DASHED line through (0, 2) with slope 1, and the region ABOVE it shaded.
Eliminate the wrong options
Which inequality does it represent? Three can be eliminated by reading the line before any substitution.
Survives elimination: b
Why: The dashed line eliminates both choices with an equal bar, leaving only two. Shading above the line means the y-values exceed those on the line, so the symbol is greater than. Confirming with the origin: is 0 greater than 2? No — and the origin does indeed sit below the line, in the unshaded half. Two free readings did most of the work here.
Trap
The trap. A point satisfies the first inequality of a system, so you select it.
A solution to a system must satisfy EVERY inequality in it. Passing one is not sufficient.
The distractors are built exactly this way: each satisfies one condition and fails another.
The fix. Substitute each candidate into every inequality, and eliminate on the first failure.
Only a point passing all of them is a solution.
Counterexample
A rule students over-generalise.
Discussion prompt
A student says: whenever a negative number appears in an inequality, flip the sign. Give a counterexample.
Hint: Try an addition involving a negative.
Answer:
Counterexample: x plus 3 is less than 1. Subtracting 3 gives x less than negative 2, with no flip, and a negative number is plainly involved.
Testing confirms it: x equals negative 5 gives negative 2, which is less than 1. Correct. The flipped version, x greater than negative 2, would include x equals 0, giving 3, which is not less than 1.
A second counterexample: negative 6 is less than x. No operation is performed at all, so nothing flips, and the statement simply reads x greater than negative 6.
The precise rule: the flip depends on the OPERATION being a multiplication or division, and on the SIGN of what you multiply or divide by. The presence of negative numbers elsewhere is irrelevant.
Why the over-generalisation is costly: it reverses correct answers, which is worse than leaving them alone, and it does so on questions the student could otherwise do.
Edge cases
Testing (0, 0) settles almost every region question. When does it fail?
Discussion prompt
When can you not use the origin as a test point, and what do you do instead?
Hint: Where must the origin not be?
Answer:
The origin fails as a test point when the boundary line passes through it — a line such as y equals 2x. Then (0, 0) sits ON the boundary rather than in either half, so substituting gives equality and settles nothing.
What to do instead: pick any other convenient point clearly off the line, such as (1, 0) or (0, 1).
How to spot it in advance: the line passes through the origin exactly when its equation has no constant term. So y equals 3x needs a different point, and y equals 3x plus 1 does not.
The method itself never fails, only that particular choice of point. Every point on one side of a line behaves identically, so any off-line point works.
A related edge: for a system, the origin may lie inside one region and outside another, which is fine — that simply tells you it is not a solution to the system.
Check
Read the line, then test the origin.
Check your understanding
A graph shows a solid line through (0, 4) with slope negative 2, with the region containing the origin shaded. Which inequality is it?
Answer: A
Why: The line is y equals negative 2x plus 4. It is solid, so the symbol includes equality, which eliminates the two strict choices. Testing the origin: is 0 less than or equal to 4? Yes — so the origin lies in the less-than-or-equal region, which is the shaded one.
Reading the line style eliminated two choices for free, and one substitution settled the remaining pair. That is the whole method for region questions.
Section
Section 5
Matching
Six phrases, six representations.
Match the pairs
Why: The bottom three rows are all the same distinction in three notations: whether the boundary itself counts. Solid line, filled circle and an equal bar all say yes; dashed line, open circle and a strict symbol all say no.
Sorting
Each is one step in solving an inequality.
Sort into buckets
Does the sign flip?
Four of six do not flip, which is worth noticing: the flip is the exception rather than the norm, and over-applying it is as costly as forgetting it.
Comparison
Fill the blanks from memory.
Comparison matrix
| linear equation | linear inequality | |
|---|---|---|
| Number of solutions | usually exactly one value | a whole range of values |
| Adding and subtracting | same on both sides | same, and never flips the sign |
| Dividing by a negative | no special rule | flips the direction of the sign |
| How to check | substitute the value | substitute one value from each side |
The bottom row is the practical difference. Checking an equation confirms a point; checking an inequality has to confirm a direction, which takes two substitutions rather than one.
Trade off
Fill in what each costs.
Comparison matrix
| route | steps | the risk |
|---|---|---|
| Divide by the negative and flip | one division, with a flip | forgetting the flip |
| Move the variable to the other side first | one extra addition, then divide by a positive | none — no flip is ever needed |
| Test the answer both sides | two substitutions | none, and it catches a missed flip |
| Trust the algebra and move on | no steps | the boundary is right and the direction is wrong |
The second row is the underused tactic: one extra addition removes the only rule that can go wrong. When the variable has a negative coefficient, moving it across is genuinely safer than remembering to flip.
Real world
One minute on why inequalities are the more useful tool.
Discussion prompt
Most real constraints are inequalities rather than equations. What does that change about the answers you get?
Answer:
Budgets, capacities, deadlines and tolerances are all limits, not targets. A lift holds at most eight people; a budget is a ceiling; a component must be within a tolerance.
So the answer is a range rather than a value, and the interesting question is usually the boundary — the most you can afford, the latest you can leave.
That is why the equal bar matters so much. Whether the limit itself is achievable is a real distinction: a bridge rated for at most 10 tonnes is different from one rated for under 10 tonnes.
Systems of inequalities are how scheduling and resource allocation actually work. Each constraint is a half-plane, and the feasible options are the overlap — exactly the region questions in this deck.
And rounding follows the context, not the decimal. 15.29 crates means 15, because the constraint is a ceiling, and that reasoning is the same in any real allocation problem.
Ranking
Solve each, then order the boundary values.
Put in order
Why: Solving each: (a) gives x less than 3, (b) multiplies by negative 1 with a flip to give x less than 5, (c) gives x less than 1, and (d) gives x less than 7. Ordering the boundaries smallest to largest: 1, 3, 5, 7. Note that (b) was the only one requiring a flip, and getting it wrong would have given x greater than 5, changing not the boundary but the direction.
Concept
This type is 4.2 per cent of the section, and it splits into a small algebra rule and a separate graphical variant. Both are quick to learn.
| session | what you do | why |
|---|---|---|
| 1 | Twenty one-variable solves, half with a negative coefficient, testing both sides of every answer. | Builds the flip habit and the two-sided check that catches it. |
| 2 | Fifteen word problems, underlining the qualifying phrase before writing any symbol. | Translation is where at most and more than are confused. |
| 3 | Fifteen region questions, reading the line style before substituting anything. | The variant students practise least, and the fastest once the routine is set. |
| 4 | Ten systems, substituting candidate points into every inequality. | Trains the eliminate-on-first-failure habit. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 71 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state exactly when the inequality sign flips, and describe how to identify a shaded region from a graph.
Hint: The first is about an operation, the second about a point.
Answer:
The sign flips only when multiplying or dividing both sides by a negative number. Adding and subtracting never flip, and multiplying or dividing by a positive never flips.
The reason: multiplying by a negative reflects the number line, so the order of any two values reverses.
For a region: read the boundary line's equation, then its style — solid means the symbol includes equality, dashed means it is strict.
Then test the point (0, 0). If it satisfies the inequality, the origin's side is the shaded one; if not, the other side is.
Use a different test point only when the line passes through the origin, which happens exactly when the equation has no constant term.
Explain it
Two minutes, out loud.
Discussion prompt
A friend either forgets the flip or applies it every time a minus sign appears. How do you give them a rule they can actually use?
Answer:
Show why it happens first. Write 2 is less than 5 on the board. Now multiply both sides by negative 1: negative 2 and negative 5. Ask which is bigger — negative 2 is. The order reversed.
Name the cause: multiplying by a negative reflects the number line, so anything that was smaller becomes larger.
Then give the rule precisely: flip only when you MULTIPLY or DIVIDE both sides by a NEGATIVE. Adding and subtracting never flip, whatever minus signs are floating around.
Give them the escape route: if the variable has a negative coefficient, move it to the other side instead. Then you divide by a positive and no flip is ever needed.
And the check: substitute one value that should work and one that should not. It catches both the missed flip and the over-applied one.
Commit first
Commit before you check.
Predict first
For the inequality y is less than 3x minus 6, is the origin in the shaded region?
Correct: no
Why: Substituting (0, 0) gives 0 less than 3 times 0 minus 6, which is 0 less than negative 6 — false. So the origin lies in the unshaded half and the shading is on the other side of the line. The origin is not on the boundary, since the line crosses the y-axis at negative 6. Line style affects only whether the boundary itself is included, not which side is shaded.
Connect it up
Blank paper.
Draw it
Split the page in two. On the left, headed ONE VARIABLE, write the solving chain identical to an equation, and beside the division step write in large letters FLIP ONLY IF NEGATIVE, with the reason: multiplying by a negative reflects the number line. Underneath, draw a number line with a filled circle and an open circle labelled with the symbols each corresponds to. On the right, headed TWO VARIABLES, draw a line with a shaded half-plane, mark the origin with a dot, and write the three-step routine: read the equation, read the line style, test (0, 0). Underneath, draw two overlapping regions and label the overlap as the solution to a system. At the bottom, list the four phrases — at most, at least, more than, fewer than — with their symbols.
Exit ticket
One question before you close the deck.
Predict first
You have solved an inequality and want to be sure the direction is right. What do you do?
Correct: Test one value from each side of the boundary in the original
Why: On this type the boundary is usually right and the direction is what fails, so checking the boundary alone confirms the half that was not at risk. Two substitutions — one value that should satisfy the inequality and one that should not — test the direction directly. Re-doing the algebra repeats whatever reasoning went wrong the first time, and confirming you flipped somewhere is not a check at all, since flipping when you should not is an error too.
Recap
One type, two rules: flip only for a negative multiplier, and test the origin for a region.
| never do this | do this instead |
|---|---|
| Carry the direction through a division by a negative | Flip at the moment you divide |
| Flip because a minus sign appears somewhere | Flip only for multiplication or division by a negative |
| Write strict less than for at most | At most includes the boundary, so use the equal bar |
| Guess which side of a line is shaded | Substitute (0, 0) and read the result |
| Accept a point that satisfies one inequality of a system | Check it against all of them |
| Check only the boundary value | Test a value on each side to confirm the direction |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
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