Linear Inequalities on the Digital SAT: Graphs, Systems & Word Problems

Linear inequalities belong to the SAT Algebra domain.

The algebra is similar to equations—with one crucial exception:

When you multiply or divide both sides by a negative number, reverse the inequality sign.

Basic example

[ 3x+4>19 ]

Subtract 4:

[ 3x>15 ]

[ x>5 ]

The steps otherwise match solving a linear equation.

Negative multiplier

[ -2x+7\le15 ]

Subtract 7:

[ -2x\le8 ]

Divide by -2 and flip:

[ x\ge-4 ]

This is one of the most common SAT inequality errors.

Compound inequalities

[ 2<x+3\le8 ]

Subtract 3 from all parts:

[ -1<x\le5 ]

This means:

• greater than -1;
• less than or equal to 5.

Graphing on a number line

• (x>3): open circle at 3, shade right
• (x\ge3): closed circle at 3, shade right
• (x<3): open circle, shade left
• (x\le3): closed circle, shade left

Word translation

Common language:

Example:

A student needs at least 75 points.

[ p\ge75 ]

Context example

A venue can hold at most 480 people. There are already 126 inside. If each arriving group contains 6 people, what is the maximum number of full groups that can enter?

[ 126+6g\le480 ]

[ 6g\le354 ]

[ g\le59 ]

Maximum full groups = 59.

Word-problem inequalities like this often appear alongside percentage problems on the SAT, so it helps to practice translating both kinds of phrasing.

Systems of inequalities

Example:

[ y>2x+1 ]

[ y\le- x+7 ]

A solution point must satisfy both.

Graphically:

• above first line;
• on/below second.

The overlap is the solution region.

Test a point

Question asks whether (2,4) satisfies:

[ y\ge x+1 ]

[ y<3x ]

Check:

[ 4\ge3 ]

true.

[ 4<6 ]

true.

So the point satisfies the system.

Parameter reasoning

If a condition says:

[ ax>12 ]

you cannot divide safely by a unless you know whether (a) is positive or negative.

The sign determines whether the inequality reverses.

That is an advanced structural trap. A similar sign-awareness trap shows up in quadratic equations when solving by factoring or completing the square.

Desmos

Desmos can graph inequalities and show shaded regions.

Useful for:

• systems;
• visual checking;
• feasible regions.

But simple one-variable inequalities are usually faster by hand. See the rest of our SAT Math guides for other Algebra topics.

Mini-practice

1

Solve:

[ 5x-7<18 ]

2

Solve:

[ -3x+4\ge19 ]

3

A club can spend no more than $900. It already spent $260. Each new kit costs $40. What is the maximum number of kits it can buy?

4

Which point satisfies both:

[ y>x ]

and

[ y\le4 ]

A. (5,3)
B. (2,4)
C. (4,4)
D. (1,0)

Answers

1. x<5
2. x≤-5
3. 16 kits
4. B

Common traps

• forgetting to flip after dividing by a negative;
• translating “at most” as ≥;
• using an open point for ≥ or ≤;
• satisfying only one inequality in a system.

Official references

• College Board Algebra: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/algebra

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