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Digital SAT · Linear Equations & Inequalities

How to Solve Linear Inequalities on the Digital SAT

Solving an inequality is just like solving an equation, with a single extra rule: multiply or divide both sides by a negative and the inequality sign flips. Isolate the variable, then read the request carefully, since these often want the greatest or least value, or the greatest integer, rather than the whole solution. Desmos can shade the solution range as a quick check on the direction.

Written from Perfect1600’s analysis of every linear inequalities question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
per test
typical difficulty
Medium
typical difficulty
practice questions
60
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

flip on ÷
One extra rule

Everything is equation-like except that a negative multiply or divide reverses the sign.

63%
Do it on Desmos

Graphing shades the solution and confirms the direction.

greatest/least
Read the request

Many want the greatest or least value, not just any solution.

How to recognize linear inequalities questions

  • An inequality symbol is present and one variable needs isolating.
  • You are asked for the greatest or least value, or the greatest integer, that works.
  • Two inequalities may be paired, asking for a value that satisfies both.
  • Choices are numbers, and most are multiple choice.

Why students miss these

Because the steps mirror an equation, the one thing people forget is flipping the sign on a negative divide. The other slip is a misread request: giving any solution instead of the greatest or least, or ignoring that a greatest-integer answer must be a whole number just under the boundary. Watch the sign as you divide, and reread what value is actually wanted.

The step-by-step method

  1. 1

    Isolate the variable

    Move and combine terms as you would for an equation.

  2. 2

    Flip on a negative

    Multiplying or dividing both sides by a negative reverses the inequality sign.

  3. 3

    Locate the boundary

    The solution is everything on one side of a boundary number; note if the boundary is included.

  4. 4

    Answer the request

    Give the greatest or least value, or the greatest integer, that the question names.

Solving it on Desmos

  1. Type the inequality. Enter it with x to see the shaded solution range.
  2. Find the edge. The boundary is the edge of the shaded region; note whether it is included.
  3. Pick the value. Read the greatest or least value, or the greatest integer, from the shaded range.

Full Desmos walkthrough for linear inequalities→

When to use it: The shaded region confirms the direction of the sign, the thing students most often flip. Solve by hand on simple ones and use the graph as a check when the negative divide worries you.
See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example
If 5x−4≤265x - 4 \leq 26, what is the greatest possible value of xx?
A
55
66
C
2222
D
3030

A: Incorrect. This ignores the −4-4 and uses 26÷526 \div 5 rounded down to 5.

B: Correct. 5x≤305x \leq 30 gives x≤6x \leq 6, so the greatest value is 6.

C: Incorrect. This forgets to divide by 5: 26−4=2226 - 4 = 22.

D: Incorrect. This forgets to divide by 5 after adding: 26+4=3026 + 4 = 30.

Explanation

Add 4 to both sides to get 5x≤305x \leq 30, then divide by 5: x≤6x \leq 6; the greatest value is 6.

Medium example
What is the greatest integer value of xx that satisfies 4x+9<304x + 9 < 30?
A
44
55
C
66
D
2121

A: Incorrect. This subtracts an extra 1 unnecessarily from the correct bound.

B: Correct. 4x<214x < 21 gives x<5.25x < 5.25, so the greatest integer is 5.

C: Incorrect. This rounds 5.25 up instead of down for a strict inequality.

D: Incorrect. This forgets to divide by 4: 30−9=2130 - 9 = 21.

Explanation

Subtract 9: 4x<214x < 21. Divide by 4: x<5.25x < 5.25. The greatest integer less than 5.25 is 5.

Hard example
y>x+2y > x + 2 and y<4x−1y < 4x - 1 Which of the following tables gives three values of x and their corresponding values of y that are all solutions to the given system of inequalities?
A
xy
20
31
42
xy
26
38
411
C
xy
26
38
41
D
xy
27
311
415

A: The pair (x,y)=(2,0)(x, y) = (2, 0) does not satisfy both inequalities in the system, so not all values in this table are solutions.

B: Every ordered pair in this table satisfies both inequalities in the system, so all three values of x and y are solutions.

C: The pair (x,y)=(4,1)(x, y) = (4, 1) does not satisfy both inequalities in the system, so not all values in this table are solutions.

D: The pair (x,y)=(2,7)(x, y) = (2, 7) does not satisfy both inequalities in the system, so not all values in this table are solutions.

Explanation

Substitute each ordered pair (x, y) from a table into both inequalities in the system. A table is correct only when all three of its pairs are solutions. The correct table is the one whose every pair makes both inequalities in the system true; any table containing even one pair that fails is incorrect.

The common traps

PatternWhat it doesThe tell
Sign not flippedDivided by a negative without reversing the inequality.Any negative multiply or divide flips the sign.
Wrong value from the rangeGave a solution instead of the greatest or least one.Check the last line again for greatest, least, or greatest integer.
Boundary in or outTreated a strict inequality as inclusive, or the reverse.A closed symbol includes the boundary; a strict one does not.

Try it: two real questions

Question 1easy

A delivery driver can carry at most 12 packages at one time and is already carrying 7 packages. What is the greatest number of additional packages the driver can carry at the same time?

Question 2easy

A food cart can serve at most 90 customers in one day. It serves bb customers in the morning and ll customers in the afternoon. Which inequality represents this situation?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Linear Equations & Inequalities
Linear Equations
Systems of Linear Equations
Inequalities from Word Problems
Linear Equations & Inequalities
Inequalities from a Graph

Related reading

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Common questions

How do you solve a linear inequality on the SAT?

Isolate the variable as in an equation, but flip the sign if you multiply or divide by a negative. Then read the boundary and give the exact value the question wants.

When do you flip the inequality sign?

Only when you multiply or divide both sides by a negative. Adding, subtracting, or dividing by a positive leaves the direction unchanged.

How do I find the greatest integer that satisfies an inequality?

Solve for the boundary, then take the largest whole number on the correct side. If x is less than 6, the greatest integer is 5, unless 6 is included.

Can Desmos solve inequalities?

Yes. Type the inequality with x and it shades the solutions, which confirms the direction and shows the boundary. Read the value you need from the shaded region.

Practice linear inequalities the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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