Digital SAT · Linear Equations & Inequalities

How to Solve Inequalities from a Graph on the Digital SAT

Q: How do you find the inequality for a shaded graph on the SAT?

Read the boundary line's slope and intercept, decide the symbol from solid or dashed, and set the direction using which side is shaded. Test a point in the shaded region to confirm.

Q: What does a dashed boundary line mean?

A dashed line means the boundary is not included, so the inequality is strict, greater than or less than. A solid line includes the boundary, so it is greater-or-equal or less-or-equal.

Q: How do I tell which side to shade?

Pick a test point not on the line, often the origin, and see if it satisfies the inequality. If it does, the region containing it is shaded; if not, the other side is.

Q: Can Desmos graph an inequality to check my answer?

Yes. Type a candidate inequality and Desmos shades its region. Compare that shading and boundary style to the picture, and confirm a shaded point satisfies it.

Here a shaded region or a boundary line is shown, and you match it to an inequality, or test which point is a solution. Read the boundary line's slope and intercept, decide the direction from which side is shaded, and note whether the line is solid or dashed. A solid line means the boundary is included, so the symbol is greater than or equal to; a dashed line means it is strict. Desmos can graph a candidate inequality to compare shading.

Written from Perfect1600’s analysis of every inequalities from a graph question in our bank·Method checked against the current Bluebook test

per test: less than 1 per test
typical difficulty: Medium
practice questions: 15

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

87%

Checkable on Desmos

Graph each candidate and match the shaded region to the picture.

100%

All multiple choice

Choices are inequalities, so testing a shaded point decides it.

33%

Rated Hard

Shading direction and dashed lines, not the boundary, cause misses.

How to recognize inequalities from a graph questions

A graph shows a shaded half-plane or a region between lines.
The question asks which inequality matches, or which ordered pair is a solution.
The boundary line is drawn solid or dashed.
Answer choices are inequalities that differ in the line or the direction.

Why students miss these

Two features have to be read correctly: the boundary and the direction of shading. Students match the slope and intercept but pick the wrong inequality direction, or miss that a dashed line excludes the boundary. Testing a point that is clearly in the shaded region against each inequality settles the direction, and checking solid versus dashed sets the symbol.

The step-by-step method

1
Read the boundary line
Find the slope and intercept of the boundary, as you would for any line.
2
Note solid or dashed
A solid line includes the boundary, so use greater-or-equal or less-or-equal; a dashed line is strict.
3
Use the shaded side
Pick a point in the shaded region and see whether y is above or below the line there.
4
Test a point to confirm
Plug the shaded point into each candidate inequality and keep the one it satisfies.

Solving it on Desmos

Graph a candidate inequality. Type an answer choice into Desmos and see the region it shades.
Compare the shading. Keep the inequality whose shaded region and boundary style match the picture.
Check a point. Confirm a point from the shaded region satisfies the inequality you chose.

Full Desmos walkthrough for inequalities from a graph→

When to use it: Desmos shades each candidate so you can match it to the graph directly, which fixes the direction and the boundary style at once. Reading the boundary by hand is enough when the line is simple.

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

In the xy-plane, a dashed line passes through

(0, -1)

with a slope of 1, and the region below the line is shaded. Which inequality represents the shaded region?

y > x - 1

y \geq x - 1

y \leq x - 1

y < x - 1

A: Incorrect. $>$ shades above the line, not below it.

B: Incorrect. $\geq$ goes with a solid line and shades above.

C: Incorrect. $\leq$ goes with a solid line, but this line is dashed.

D: Correct. The boundary is $y = x - 1$ ; a dashed line with the region below shaded gives $y < x - 1$ .

Explanation

The boundary is $y = x - 1$ . Dashed means strict ( $<$ or $>$ ), and shading below gives $y < x - 1$ .

Medium example

Which ordered pair

(x, y)

is a solution to the inequality

y - 2x \geq 1

(0, 0)

(1, 1)

(2, 6)

(3, 2)

A: Incorrect. At $(0, 0)$ , $0 - 2(0) = 0$ , which is not at least 1.

B: Incorrect. At $(1, 1)$ , $1 - 2(1) = -1$ , which is not at least 1.

C: Correct. At $(2, 6)$ , $6 - 2(2) = 2 \geq 1$ is true.

D: Incorrect. At $(3, 2)$ , $2 - 2(3) = -4$ , which is not at least 1.

Explanation

Substitute each pair into $y - 2x \geq 1$ . Only $(2, 6)$ gives a true statement: $2 \geq 1$ .

Hard example

In the xy-plane, the boundary of the region described by

y \geq kx - 4

is a solid line that passes through the point

(2, 2)

. What is the value of

k

-1

1

3

6

A: Incorrect. This computes $\frac{2 - 4}{2}$ with a sign error on the constant.

B: Incorrect. This ignores the $-4$ and uses $\frac{2}{2}$ .

C: Correct. The boundary $y = kx - 4$ passes through $(2, 2)$ : $2 = 2k - 4$ , so $k = 3$ .

D: Incorrect. This stops at $2k = 6$ without dividing by 2.

Explanation

The boundary line is $y = kx - 4$ . Substituting $(2, 2)$ : $2 = 2k - 4$ , so $2k = 6$ and $k = 3$ .

Detailed explanation

The boundary of the region is the line $y = kx - 4$ . Since it passes through $(2, 2)$ , substitute to get $2 = 2k - 4$ . Adding 4 gives $2k = 6$ , so $k = 3$ .

The common traps

Pattern	What it does	The tell
Wrong shading direction	Chose the opposite inequality symbol for the shaded side.	Test a point in the shaded region; it must satisfy the inequality.
Solid versus dashed	Used a strict symbol for a solid line, or the reverse.	Solid includes the boundary; dashed excludes it.
Boundary misread	Got the slope or intercept of the boundary wrong.	Read the line as you would any line before choosing the direction.

Try it: two real questions

Question 1easy

In the xy-plane, a solid line passes through $(0, 3)$ with a slope of 2, and the region above the line is shaded. Which inequality represents the shaded region?

Question 2easy

Which ordered pair $(x, y)$ is a solution to the inequality $y < x + 2$ ?

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

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Linear Functions

Slope of a Line

Common questions

How do you find the inequality for a shaded graph on the SAT?

Read the boundary line's slope and intercept, decide the symbol from solid or dashed, and set the direction using which side is shaded. Test a point in the shaded region to confirm.

What does a dashed boundary line mean?

A dashed line means the boundary is not included, so the inequality is strict, greater than or less than. A solid line includes the boundary, so it is greater-or-equal or less-or-equal.

How do I tell which side to shade?

Pick a test point not on the line, often the origin, and see if it satisfies the inequality. If it does, the region containing it is shaded; if not, the other side is.

Can Desmos graph an inequality to check my answer?

Yes. Type a candidate inequality and Desmos shades its region. Compare that shading and boundary style to the picture, and confirm a shaded point satisfies it.

Practice inequalities from a graph 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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What the question bank shows

How to recognize inequalities from a graph questions

Why students miss these

The step-by-step method

Read the boundary line

Note solid or dashed

Use the shaded side

Test a point to confirm

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you find the inequality for a shaded graph on the SAT?

What does a dashed boundary line mean?

How do I tell which side to shade?

Can Desmos graph an inequality to check my answer?

Practice inequalities from a graph the way it is tested