Digital SAT · Quadratic & Exponential Functions

How to Solve Absolute Value Equations on the Digital SAT

Absolute value is distance from zero, so two numbers, one positive and one negative, usually satisfy it. That means an equation like the absolute value of A equals b breaks into two: A equals b, or A equals negative b. Before you solve either, glance at the right side, because a distance can never equal a negative, and that alone means no solution. On the digital test you can also graph both sides and count where they cross.

Written from Perfect1600’s analysis of every absolute value equations question in our bank·Method checked against the current Bluebook test

per test: less than 1 per test
typical difficulty: Mostly hard
practice questions: 42

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

Two cases, one check

Inside equals the value or its negative; a negative right side means none.

90%

Do it on Desmos

Graph both sides and count crossings for the solutions.

50%

Grid-ins

Half are typed answers, so a forgotten second case has nothing to catch it.

How to recognize absolute value equations questions

Absolute value bars wrap an expression set equal to a number.
You are asked for a solution, the positive or negative one, or the number of solutions.
The right side might be negative, a hint at no solution.
Choices are numbers or a count of solutions.

Why students miss these

The second case is easy to forget, so a student reports the positive answer and stops. Others solve away when the right side is negative and no solution exists. The count version hides a small rule: a right side of zero gives one solution, a positive right side gives two, a negative right side gives none. Splitting into two cases and glancing at the right side first covers every version.

The step-by-step method

1
Glance at the right side
If the absolute value equals a negative number, stop: there is no solution.
2
Split into two cases
Set the inside equal to the value and to its negative: A equals b, or A equals negative b.
3
Solve both
Work each equation to get both solutions.
4
Give what is asked
The positive solution, the negative one, both, or a count, per the question.

Solving it on Desmos

Graph both sides. Type y = the absolute value expression and y = the number into Desmos.
Count the crossings. Two for a positive right side, one for zero, none for a negative.
Read the solutions. Click each crossing for its x-value, then answer the specific one wanted.

Full Desmos walkthrough for absolute value equations→

When to use it: 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.

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

What is the positive solution to the equation

|x + 2| = 6

-8

-2

4

6

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

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

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

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

Explanation

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

Medium example

What is the negative solution to the equation

|2x + 1| = 9

-5

4

5

8

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

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

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

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

Explanation

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

Hard example

How many distinct real solutions does the equation

|x - 3| = -2

have?

Exactly one

Exactly two

Zero

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 $-2$ is negative.

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

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

Explanation

Since $|x - 3| \geq 0$ for all $x$ , it can never equal $-2$ ; 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 $-2$ , no value of $x$ works, giving zero solutions. Any positive right side would instead give two solutions, and a right side of 0 would give one.

The common traps

Pattern	What it does	The tell
Only the first case	Reported the positive solution and skipped the negative.	Absolute value gives two cases unless the right side is zero.
Solved past no solution	Worked an equation set equal to a negative number.	A negative right side means no solution.
Miscounted	Gave the wrong number on a count version.	Positive gives two, zero gives one, negative gives none.

Try it: two real questions

Question 1easy

Graph

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

Question 2easy

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

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.

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Often confused with

Linear Equations & Inequalities

Linear Equations

Quadratic & Exponential Functions

Quadratic Equations

Linear Equations & Inequalities

Linear Inequalities

Common questions

How do you solve an absolute value equation on the SAT?

Set the inside equal to the value and to its negative, then solve both. First glance at the right side: a negative there means no solution, since distance is never negative.

Why do absolute value equations usually have two solutions?

Because absolute value measures distance from zero, and two numbers sit the same distance away, one on each side. So the inside can equal the value or its opposite.

When does an absolute value equation have no solution?

When it equals a negative number. Absolute value is never negative, so it cannot equal one, and there is no solution.

Can Desmos solve absolute value equations?

Yes. Graph y = the absolute value expression and y = the number, then count the crossings and click each for its value. Two crossings, one, or none.

Practice absolute value equations 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 absolute value equations questions

Why students miss these

The step-by-step method

Glance at the right side

Split into two cases

Solve both

Give what is asked

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 solve an absolute value equation on the SAT?

Why do absolute value equations usually have two solutions?

When does an absolute value equation have no solution?

Can Desmos solve absolute value equations?

Practice absolute value equations the way it is tested