Algebra · SAT Math

Linear Inequalities on the Digital SAT

Linear inequalities work exactly like linear equations — with one critical twist that catches students every time. This lesson covers the basic solving technique, the flip-sign rule, compound inequalities, graphing solution sets, and systems of inequalities. Plus 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~9 min read

Solving inequalities (mostly) like equations

A linear inequality is a linear equation with <, >, ≤, or ≥ instead of =. You solve it almost exactly the same way: use inverse operations to isolate the variable.

Example. Solve 3x + 4 > 19.

Subtract 4: 3x > 15.

Divide by 3: x > 5. (Solution: every x greater than 5.)

The solution is not a single number — it's a range. x > 5 includes 5.0001, 6, 7, 100, etc.

The flip-sign rule (the one twist)

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

Example. Solve -2x > 8.

Divide both sides by -2, AND flip > to <: x < -4.

Why? Multiplying by a negative reverses the order on the number line. 3 > 1, but -3 < -1. The inequality sign reflects the new ordering.

Important: you don't flip when adding or subtracting negatives, or when dividing by a positive — only when multiplying or dividing both sides by a negative number.

Compound inequalities

A compound inequality has the variable trapped between two bounds, like 5 < 2x + 1 < 11. To solve, perform the same operation on all three parts simultaneously.

Example. Solve 5 < 2x + 1 < 11.

Subtract 1 from all three: 4 < 2x < 10.

Divide by 2: 2 < x < 5.

Solution: any x strictly between 2 and 5.

Graphing and systems of inequalities

Inequalities in two variables (like y < 2x + 3) graph as a half-plane bounded by a line. Conventions:

  • Solid line for ≤ or ≥ (the line is included).
  • Dashed line for < or > (the line is excluded).
  • Shade above the line if y > ... or y ≥ ....
  • Shade below the line if y < ... or y ≤ ....

For a system of inequalities, graph each one and the solution is the region where the shadings overlap. Test a point (like (0, 0)) to verify it satisfies both inequalities.

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Solving with Desmos

Desmos handles inequalities natively. Type them as written and Desmos shades the solution region. Three techniques:

1. Graph the inequality directly

Type y > 2x + 3 and Desmos shades the half-plane above the line, with a dashed boundary (since >, not ≥).

2. System of inequalities — find the overlap

Enter both inequalities on separate lines. Desmos shows the shaded region for each in different colors. The overlap is the solution region — usually a darker shade.

3. Check if a point is a solution

Type the point as (3, 4). Desmos plots it. If it's inside the shaded region of an inequality, it's a solution.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Forgetting to flip the sign with a negative

When you divide -3x > 12 by -3, the inequality flips: x < -4. Most students remember this on the first try and forget it on the third. Build the habit.

Flipping when you shouldn't

You only flip when multiplying or dividing by a negative — NOT when adding or subtracting a negative. 5 - 3 > 2 - 3 stays 2 > -1.

Confusing < with ≤ on the boundary

x < 5 does NOT include 5. x ≤ 5 includes 5. The SAT loves to put 5 in the answer choices to test this.

Shading the wrong side of the line

For y > [line], shade ABOVE the line, not below. Quick check: pick a test point like (0, 0), plug into the inequality. If the test point satisfies it, shade that side.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

If 5x - 3 < 12, what is the largest integer value of x?

  1. 1
  2. 2
  3. 3
  4. 4

Click to reveal solution →

Answer: (B) 2

Add 3: 5x < 15. Divide by 5: x < 3.

Since x must be strictly less than 3, the largest integer is 2.

2

Solve -3x + 7 ≥ 19.

  1. x ≤ -4
  2. x ≥ -4
  3. x ≤ 4
  4. x ≥ 4

Click to reveal solution →

Answer: (A) x ≤ -4

Subtract 7: -3x ≥ 12.

Divide by -3 AND flip the sign: x ≤ -4.

3

Which value of x is NOT a solution to 2 < 3x - 1 ≤ 14?

  1. 1
  2. 2
  3. 4
  4. 5

Click to reveal solution →

Answer: (A) 1

Solve: add 1 to all parts: 3 < 3x ≤ 15. Divide by 3: 1 < x ≤ 5.

x = 1 is NOT included (strict inequality). The others are.

4

Maria has at most $60 to spend on apples and oranges. Apples cost $1.50 each and oranges cost $2 each. If she buys a apples and r oranges, which inequality represents her budget?

  1. 1.50a + 2r &lt; 60
  2. 1.50a + 2r ≤ 60
  3. 1.50a + 2r ≥ 60
  4. 1.50a + 2r = 60

Click to reveal solution →

Answer: (B) 1.50a + 2r ≤ 60

"At most $60" means she can spend up to and including $60. That's ≤ 60, not < 60.

5

If 4(x - 2) > 6x - 14, what is the solution?

  1. x &lt; 3
  2. x &gt; 3
  3. x &lt; -3
  4. x &gt; -3

Click to reveal solution →

Answer: (A) x < 3

Distribute: 4x - 8 > 6x - 14.

Subtract 4x: -8 > 2x - 14. Add 14: 6 > 2x.

Divide by 2 (positive — don't flip): 3 > x, i.e., x < 3.

6

For the inequality y < 2x + 1, is the point (0, 0) a solution?

  1. Yes
  2. No
  3. Only if x ≥ 0
  4. Only if x &lt; 0

Click to reveal solution →

Answer: (A) Yes

Plug in: 0 < 2(0) + 1 → 0 < 1. True. So (0, 0) satisfies the inequality.

7

A movie streaming service charges a $5 monthly fee plus $2.50 per movie. If a member wants to spend no more than $30 in a month, what is the maximum number of movies they can watch?

  1. 8
  2. 9
  3. 10
  4. 11

Click to reveal solution →

Answer: (C) 10

Let m = movies. Inequality: 5 + 2.50m ≤ 30.

Subtract 5: 2.50m ≤ 25. Divide by 2.50: m ≤ 10.

Maximum integer is m = 10.

8

Which point (x, y) is in the solution region of the system y ≥ x + 1 and y < 2x + 3?

  1. (0, 0)
  2. (0, 4)
  3. (2, 4)
  4. (2, 8)

Click to reveal solution →

Answer: (C) (2, 4)

Check (2, 4) in both: 4 ≥ 2 + 1 = 3 ✓ and 4 < 2(2) + 3 = 7 ✓. Both satisfied.

(Other choices fail at least one inequality — verify by plugging in.)

Frequently asked questions

What is a linear inequality?

+
A linear inequality is like a linear equation but with <, >, ≤, or ≥ instead of =. The solution is a range of values, not a single number — for example, x > 5 means every number greater than 5.

When do I flip the inequality sign?

+
You flip the sign when you multiply or divide both sides by a negative number. You do NOT flip when adding, subtracting, or dividing by a positive.

What's the difference between < and ≤?

+
< (strict) does not include the boundary value. ≤ (inclusive) does. For x ≤ 5, x = 5 is allowed; for x < 5, it isn't.

How do I graph a linear inequality in two variables?

+
Graph the boundary line (solid for ≤ or ≥, dashed for < or >), then shade the half-plane that contains the solutions. Test a point like (0, 0) — if it satisfies the inequality, shade that side.

How do I solve a system of inequalities?

+
Graph each inequality individually, then find the overlapping region. Every point in the overlap satisfies all the inequalities at once. Test points from your answer choices in each inequality to verify.

Keep going

Solving linear equations

Same rules, equations only

Systems of equations

When two equations share variables

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