Solve the Inequality for x (x-4)^2<=0

${(x - 4)}^{2} \leq 0$

Step 1

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(x - 4)}^{2}} \leq \sqrt{0}$

Step 2

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x - 4 | \leq \sqrt{0}$

Simplify the right side.

Simplify $\sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$| x - 4 | \leq \sqrt{0^{2}}$

Pull terms out from under the radical.

$| x - 4 | \leq | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x - 4 | \leq 0$

Step 3

Write $| x - 4 | \leq 0$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x - 4 \geq 0$

Add $4$ to both sides of the inequality.

$x \geq 4$

In the piece where $x - 4$ is non-negative, remove the absolute value.

$x - 4 \leq 0$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x - 4 < 0$

Add $4$ to both sides of the inequality.

$x < 4$

In the piece where $x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 4) \leq 0$

Write as a piecewise.

${\begin{cases} x - 4 \leq 0 & x \geq 4 \\ - (x - 4) \leq 0 & x < 4 \end{cases}$

Simplify $- (x - 4) \leq 0$ .

Apply the distributive property.

${\begin{cases} x - 4 \leq 0 & x \geq 4 \\ - x - - 4 \leq 0 & x < 4 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} x - 4 \leq 0 & x \geq 4 \\ - x + 4 \leq 0 & x < 4 \end{cases}$

Step 4

Solve $x - 4 \leq 0$ when $x \geq 4$ .

Add $4$ to both sides of the inequality.

$x \leq 4$

Find the intersection of $x \leq 4$ and $x \geq 4$ .

$x = 4$

Step 5

Solve $- x + 4 \leq 0$ when $x < 4$ .

Solve $- x + 4 \leq 0$ for $x$ .

Subtract $4$ from both sides of the inequality.

$- x \leq - 4$

Divide each term in $- x \leq - 4$ by $- 1$ and simplify.

Divide each term in $- x \leq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x \geq 4$

Find the intersection of $x \geq 4$ and $x < 4$ .

No solution

Step 6

Find the union of the solutions.

$x = 4$

Step 7

Solve the Inequality for x (x-4)^2<=0

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