Solved on Dec 12, 2023

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

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

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

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

STEP 1

Assumptions1. We are looking for real numbers that satisfy the equation ∣x∣=x |x| = x ∣x∣=x.2. The absolute value function ∣x∣ |x| ∣x∣ outputs the non-negative value of x x x for any real number x x x.

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

Solve the equation −x=x -x = x −x=x for x x x.Adding x x x to both sides of the equation, we get: −x+x=x+x0=2x -x + x = x + x \\ 0 = 2x −x+x=x+x0=2x

STEP 8

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

STEP 9

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

STEP 10

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

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 $x$ .

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

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.

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 $|x| = x$ .

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

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

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

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

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