Digital SAT · Solve it in Desmos

How to Solve Inequalities from a Graph on Desmos (Digital SAT)

Q: Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Q: Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

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.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for inequalities from a graph

The Desmos steps

1
Graph a candidate inequality
Type an answer choice into Desmos and see the region it shades.
2
Compare the shading
Keep the inequality whose shaded region and boundary style match the picture.
3
Check a point
Confirm a point from the shaded region satisfies the inequality you chose.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

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 $k = 3$ .

Try it with the calculator

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.

Start the free diagnostic

Desmos questions

Can Desmos solve inequalities from a graph questions on the SAT?

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.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master inequalities from a graph, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full method Take the free diagnostic

The Desmos steps

Graph a candidate inequality

Compare the shading

Check a point

Worked in Desmos

Try it with the calculator

Question 3 is ready when you are

Desmos questions

Can Desmos solve inequalities from a graph questions on the SAT?

Is the Desmos calculator really built into the Digital SAT?

Should I still learn the algebra?

Master inequalities from a graph, calculator and all