Solve the Absolute Value Inequality for x 5|x-7|+8>=48

$5 | x - 7 | + 8 \geq 48$

Step 1

Write $5 | x - 7 | + 8 \geq 48$ as a piecewise.

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

$x - 7 \geq 0$

Add $7$ to both sides of the inequality.

$x \geq 7$

In the piece where $x - 7$ is non-negative, remove the absolute value.

$5 (x - 7) + 8 \geq 48$

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

$x - 7 < 0$

Add $7$ to both sides of the inequality.

$x < 7$

In the piece where $x - 7$ is negative, remove the absolute value and multiply by $- 1$ .

$5 (- (x - 7)) + 8 \geq 48$

Write as a piecewise.

${\begin{cases} 5 (x - 7) + 8 \geq 48 & x \geq 7 \\ 5 (- (x - 7)) + 8 \geq 48 & x < 7 \end{cases}$

Simplify $5 (x - 7) + 8 \geq 48$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x + 5 \cdot - 7 + 8 \geq 48 & x \geq 7 \\ 5 (- (x - 7)) + 8 \geq 48 & x < 7 \end{cases}$

Multiply $5$ by $- 7$ .

${\begin{cases} 5 x - 35 + 8 \geq 48 & x \geq 7 \\ 5 (- (x - 7)) + 8 \geq 48 & x < 7 \end{cases}$

Add $- 35$ and $8$ .

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ 5 (- (x - 7)) + 8 \geq 48 & x < 7 \end{cases}$

Simplify $5 (- (x - 7)) + 8 \geq 48$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ 5 (- x - - 7) + 8 \geq 48 & x < 7 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ 5 (- x + 7) + 8 \geq 48 & x < 7 \end{cases}$

Apply the distributive property.

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ 5 (- x) + 5 \cdot 7 + 8 \geq 48 & x < 7 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ - 5 x + 5 \cdot 7 + 8 \geq 48 & x < 7 \end{cases}$

Multiply $5$ by $7$ .

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ - 5 x + 35 + 8 \geq 48 & x < 7 \end{cases}$

Add $35$ and $8$ .

${\begin{cases} 5 x - 27 \geq 48 & x \geq 7 \\ - 5 x + 43 \geq 48 & x < 7 \end{cases}$

Step 2

Solve $5 x - 27 \geq 48$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Add $27$ to both sides of the inequality.

$5 x \geq 48 + 27$

Add $48$ and $27$ .

$5 x \geq 75$

Divide each term in $5 x \geq 75$ by $5$ and simplify.

Divide each term in $5 x \geq 75$ by $5$ .

$\frac{5 x}{5} \geq \frac{75}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{75}{5}$

Divide $x$ by $1$ .

$x \geq \frac{75}{5}$

Simplify the right side.

Divide $75$ by $5$ .

$x \geq 15$

Step 3

Solve $- 5 x + 43 \geq 48$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $43$ from both sides of the inequality.

$- 5 x \geq 48 - 43$

Subtract $43$ from $48$ .

$- 5 x \geq 5$

Divide each term in $- 5 x \geq 5$ by $- 5$ and simplify.

Divide each term in $- 5 x \geq 5$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \leq \frac{5}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} \leq \frac{5}{- 5}$

Divide $x$ by $1$ .

$x \leq \frac{5}{- 5}$

Simplify the right side.

Divide $5$ by $- 5$ .

$x \leq - 1$

Step 4

Find the union of the solutions.

$x \leq - 1$ or $x \geq 15$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 1 or x \geq 15$

Interval Notation:

$(- \infty, - 1] \cup [15, \infty)$

Step 6

Solve the Absolute Value Inequality for x 5|x-7|+8>=48

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