SAT Math skill page

SAT Absolute Value Equations Practice

Q: Why can an absolute-value equation have two solutions?

Absolute value measures distance from zero, so a positive distance can occur on either side of zero.

Q: When does an absolute-value equation have no solution?

It has no solution when an isolated absolute value is set equal to a negative number.

Turn absolute value statements into distance relationships and check both possible cases.

Practice absolute value equations in Dolphin SAT Practice questions

10-16 min practice time2 questions on pageAdvanced Math

Practice time10-16 min

Question bank2 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Solve simple absolute value equations
Set up two cases from an absolute value statement
Identify when an absolute value equation has no solution
Check proposed solutions in the original equation

Dolphin question bank

Practice your skill

Question 1 of 2Hard

What is the sum of all solutions to the equation |

x^{2}

- 4| = 3x?

Show answer and solution

Correct answerB. 5

Solution walkthrough

To solve $∣ x^{2} - 4∣ = 3 x$ , consider two cases: Case 1: $x^{2} - 4 = 3 x$ . Rearrange to get $x^{2} - 3 x - 4 = 0$ . Factor to $(x - 4) (x + 1) = 0$ .
Thus, $x = 4$ or $x = - 1$ . Case 2: $- (x^{2} - 4) = 3 x$ . This simplifies to $x^{2} + 3 x - 4 = 0$ .
Factor to $(x + 4) (x - 1) = 0$ . Thus, $x = - 4$ or $x = 1$ . Check each solution in the original equation: - For $x = 4$ : $∣16 - 4∣ = 3 (4)$ , true. - For $x = - 1$ : $∣1 - 4∣ = 3 (- 1)$ , false. - For $x = - 4$ : $∣16 - 4∣ = 3 (- 4)$ , false. - For $x = 1$ : $∣1 - 4∣ = 3 (1)$ , true.
Valid solutions are $x = 4$ and $x = 1$ . The sum is $4 + 1 = 5$ .

Question 2 of 2Easy

What is one possible solution to the given equation? |x - 2| = 9

Show answer and solution

Correct answerA. 11

Solution walkthrough

The correct answer is 11 or -7. By the definition of absolute value, if $∣ x - 2∣ = 9$ , then $x - 2 = 9$ or $x - 2 = - 9$ .
Adding 2 to both sides of the equation $x - 2 = 9$ yields $x = 11$ . Adding 2 to both sides of the equation $x - 2 = - 9$ yields $x = - 7$ .
Thus, the given equation, $∣ x - 2∣ = 9$ , has two possible solutions, 11 and -7. Note that 11 and -7 are examples of ways to enter a correct answer.

Avoid these traps

Common mistakes on this skill

Solving only the positive case

If |expression| = k with k > 0, the expression can equal k or -k.

Keeping solutions when the absolute value equals a negative number

An absolute value cannot be negative, so |expression| = k has no solution when k < 0.

Study plan

How to practice this skill in Dolphin

Isolate the absolute-value expression before splitting into cases.
Write and solve both equations when the other side is positive.
Substitute every candidate into the original equation and apply any domain restrictions.

Practice absolute value equations in Dolphin SAT

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FAQ

Questions about SAT Absolute Value Equations Practice

Why can an absolute-value equation have two solutions?

Absolute value measures distance from zero, so a positive distance can occur on either side of zero.

When does an absolute-value equation have no solution?

It has no solution when an isolated absolute value is set equal to a negative number.