Question Find the real numbers $x$ that satisfy the equation $|x| = x$ . Express the solution set in set-builder notation.

Studdy Solution

STEP 1

Assumptions
1. We are looking for real numbers that satisfy the equation $|x| = x$ .
2. The absolute value function $|x|$ outputs the non-negative value of $x$ for any real number $x$ .

STEP 2

Understand the definition of absolute value.
The absolute value of $x$ , denoted $|x|$ , is defined as: $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

STEP 3

Analyze the equation $|x| = x$ for the case when $x \geq 0$ .
If $x \geq 0$ , then $|x| = x$ by the definition of absolute value.

STEP 4

Conclude that all non-negative real numbers satisfy the equation when $x \geq 0$ .
Therefore, any $x$ such that $x \geq 0$ is a solution to the equation $|x| = x$ .

STEP 5

Analyze the equation $|x| = x$ for the case when $x < 0$ .
If $x < 0$ , then $|x| = -x$ by the definition of absolute value.

STEP 6

Substitute $-x$ for $|x|$ in the equation $|x| = x$ when $x < 0$ .
We get $-x = x$ .

STEP 7

Solve the equation $-x = x$ for $x$ .
Adding $x$ to both sides of the equation, we get: $-x + x = x + x \\ 0 = 2x$

STEP 8

Divide both sides of the equation $0 = 2x$ by 2 to solve for $x$ .
We get $x = 0$ .

STEP 9

Note that $x = 0$ is included in the non-negative real numbers, so we do not need to consider it as a separate case.

STEP 10

Combine the results from STEP_4 and STEP_9 to write the solution in set-builder notation.
The solution set is: $\{ x \in \mathbb{R} \mid x \geq 0 \}$
This is the set of all non-negative real numbers.
The real numbers that satisfy the equation $|x| = x$ are all non-negative real numbers, including zero.

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