Perfect1600

Digital SAT · Solve it in Desmos

How to Solve Absolute Value Equations on Desmos (Digital SAT)

Graphing makes both the solutions and the count obvious, since you just count crossings. Split into two cases by hand when the numbers are clean and you want exact values fast.

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

Full method for absolute value equations

The Desmos steps

  1. 1

    Graph both sides

    Type y = the absolute value expression and y = the number into Desmos.

  2. 2

    Count the crossings

    Two for a positive right side, one for zero, none for a negative.

  3. 3

    Read the solutions

    Click each crossing for its x-value, then answer the specific one wanted.

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

Worked in Desmos

Easy example
What is the positive solution to the equation ∣x+2∣=6|x + 2| = 6?
A
−8-8
B
−2-2
44
D
66

A: Incorrect. This is the negative solution, from x+2=−6x + 2 = -6, not the positive one.

B: Incorrect. This negates the constant inside the absolute value instead of solving.

C: Correct. The case x+2=6x + 2 = 6 gives x=4x = 4, the positive solution.

D: Incorrect. This is the value on the right side, not a solution of the equation.

Explanation

Split into x+2=6x + 2 = 6 and x+2=−6x + 2 = -6, giving 4 and −8-8; the positive one is 4.

Medium example
What is the negative solution to the equation ∣2x+1∣=9|2x + 1| = 9?
−5-5
B
44
C
55
D
88

A: Correct. The case 2x+1=−92x + 1 = -9 gives 2x=−102x = -10, so x=−5x = -5.

B: Incorrect. This is the positive solution, from 2x+1=92x + 1 = 9, not the negative one.

C: Incorrect. This adds 1 instead of subtracting before dividing by 2.

D: Incorrect. This solves 2x=82x = 8 but forgets to divide by 2.

Explanation

Split into 2x+1=92x + 1 = 9 and 2x+1=−92x + 1 = -9, giving 4 and −5-5; the negative one is −5-5.

Hard example
How many distinct real solutions does the equation ∣x−3∣=−2|x - 3| = -2 have?
A
Exactly one
B
Exactly two
Zero
D
Infinitely many

A: Incorrect. A single solution would require the right side to be 0, but here it is negative.

B: Incorrect. Two solutions would require a positive right side, but −2-2 is negative.

C: Correct. An absolute value is never negative, so ∣x−3∣=−2|x - 3| = -2 has no solution.

D: Incorrect. Infinitely many solutions would require an identity, which this is not.

Explanation

Since ∣x−3∣≥0|x - 3| \geq 0 for all xx, it can never equal −2-2; there are zero solutions.

Detailed explanation

The absolute value of any expression is greater than or equal to 0, so it can never equal a negative number. Because the right side is −2-2, no value of xx works, giving zero solutions. Any positive right side would instead give two solutions, and a right side of 0 would give one.

Try it with the calculator

Question 1easy
Graph
-15-10-552468101214Oxy

What is the lesser of the two solutions to the equation ∣x+5∣=12|x + 5| = 12?

Question 2easy

What is the positive solution to the equation ∣x−3∣=10|x - 3| = 10?

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 absolute value equations questions on the SAT?

Graphing makes both the solutions and the count obvious, since you just count crossings. Split into two cases by hand when the numbers are clean and you want exact values fast.

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 absolute value equations, calculator and all

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

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