Question type

Absolute value equations on the SAT

Q: How many solutions can an absolute value equation have?

Two when the absolute value equals a positive number, one when it equals zero, and none when it equals a negative number.

Q: How do you find when |x + 3| = k has exactly one solution?

Exactly one solution happens when the absolute value equals zero, so k = 0. If there are other terms, isolate the absolute value first.

Split $∣ A ∣ = B$ into two equations, $A = B$ and $A = - B$ , and solve both. There is no solution when $B$ is negative, exactly one when $B = 0$ , and two when $B$ is positive. Isolate the absolute value first: $2∣ x - 4∣ + 9 = k$ has exactly one solution when $k = 9$ .

These questions ask you to solve an absolute value equation, count or add its solutions, or find the constant that leaves exactly one solution.

It appeared on two of the five official practice tests we analysed. It is one of the few question types where graphing is not the safe shortcut.

On 2 of 5 official practice tests
Desmos misleads

Why students miss it

Solving only one case. $∣2 x - 5∣ = 9$ has two solutions, 7 and $- 2$ , and a question about their sum needs both.

Splitting before isolating. In $2∣ x - 4∣ + 9 = k$ , move the 9 and divide by 2 first; the "exactly one solution" case is when the absolute value equals 0, which is $k = 9$ , not $k = 0$ .

Forgetting that an absolute value is never negative: $∣ x - 4∣ = - 7$ has no solution at all.

How to spot it

Vertical bars around an expression: $∣2 x - 5∣$ .
"How many solutions", "the sum of the solutions", or "exactly one solution".
A constant on the other side, such as $k$ , that you must choose.

The method

Two cases

Get the absolute value alone on one side: $∣ A ∣ = B$ .

$B > 0$: two solutions: solve $A = B$ and $A = - B$ .
$B = 0$: exactly one solution: $A = 0$ .
$B < 0$: no solution.

With Desmos

Desmos graphs absolute value fine, but the case the SAT asks about most is the one it shows worst.

Graph y=|2x-5| and y=9: two clear intersection points, which Desmos labels.
For "exactly one solution", the line touches the corner of the V at a single point, and Desmos does not always mark that point.

Use the two cases for any question about how many solutions there are. Use Desmos only to check values you already found.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#374Adv

What is the sum of the solutions to the equation |2x − 5| = 9 ?

Pick an answer, then check it.

Show the answer and explanation

Split the absolute value:

2 x - 5 = 9

gives

x = 7

, and

2 x - 5 = - 9

gives

x = - 2

. Their sum is

7 + (- 2) = 5

2Question 2 of 2#366Adv

In the equation 2|x − 4| + 9 = k, k is a constant. If the equation has exactly one solution, what is the value of k?

Pick an answer, then check it.

Show the answer and explanation

An absolute value equation like this has exactly one solution only when the absolute value part is forced to be

0

— otherwise you get two symmetric solutions (or none). So you need

2∣ x - 4∣ = 0

, meaning

k

must equal the leftover constant:

k = 9

. At

k = 9

x = 4

is the only solution.

Questions students ask

How many solutions can an absolute value equation have?