Solve for x 4|1-2x|-7>=21

$4 | 1 - 2 x | - 7 \geq 21$

Write $4 | 1 - 2 x | - 7 \geq 21$ as a piecewise.

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

$1 - 2 x \geq 0$

Solve the inequality.

Subtract $1$ from both sides of the inequality.

$- 2 x \geq - 1$

Divide each term in $- 2 x \geq - 1$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq - 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 1}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 1}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{- 1}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{1}{2}$

In the piece where $1 - 2 x$ is non-negative, remove the absolute value.

$4 (1 - 2 x) - 7 \geq 21$

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

$1 - 2 x < 0$

Solve the inequality.

Subtract $1$ from both sides of the inequality.

$- 2 x < - 1$

Divide each term in $- 2 x < - 1$ by $- 2$ and simplify.

Divide each term in $- 2 x < - 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{- 1}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{- 1}{- 2}$

Divide $x$ by $1$ .

$x > \frac{- 1}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{1}{2}$

In the piece where $1 - 2 x$ is negative, remove the absolute value and multiply by $- 1$ .

$4 (- (1 - 2 x)) - 7 \geq 21$

Write as a piecewise.

${\begin{cases} 4 (1 - 2 x) - 7 \geq 21 & x \leq \frac{1}{2} \\ 4 (- (1 - 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Simplify $4 (1 - 2 x) - 7 \geq 21$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 \cdot 1 + 4 (- 2 x) - 7 \geq 21 & x \leq \frac{1}{2} \\ 4 (- (1 - 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Multiply $4$ by $1$ .

${\begin{cases} 4 + 4 (- 2 x) - 7 \geq 21 & x \leq \frac{1}{2} \\ 4 (- (1 - 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Multiply $- 2$ by $4$ .

${\begin{cases} 4 - 8 x - 7 \geq 21 & x \leq \frac{1}{2} \\ 4 (- (1 - 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Subtract $7$ from $4$ .

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ 4 (- (1 - 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Simplify $4 (- (1 - 2 x)) - 7 \geq 21$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ 4 (- 1 \cdot 1 - (- 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ 4 (- 1 - (- 2 x)) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ 4 (- 1 + 2 x) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ 4 \cdot - 1 + 4 (2 x) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ - 4 + 4 (2 x) - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Multiply $2$ by $4$ .

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ - 4 + 8 x - 7 \geq 21 & x > \frac{1}{2} \end{cases}$

Subtract $7$ from $- 4$ .

${\begin{cases} - 8 x - 3 \geq 21 & x \leq \frac{1}{2} \\ 8 x - 11 \geq 21 & x > \frac{1}{2} \end{cases}$

Solve $- 8 x - 3 \geq 21$ for $x$ .

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

Add $3$ to both sides of the inequality.

$- 8 x \geq 21 + 3$

Add $21$ and $3$ .

$- 8 x \geq 24$

Divide each term in $- 8 x \geq 24$ by $- 8$ and simplify.

Divide each term in $- 8 x \geq 24$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{24}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{24}{- 8}$

Divide $x$ by $1$ .

$x \leq \frac{24}{- 8}$

Simplify the right side.

Divide $24$ by $- 8$ .

$x \leq - 3$

Solve $8 x - 11 \geq 21$ for $x$ .

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

Add $11$ to both sides of the inequality.

$8 x \geq 21 + 11$

Add $21$ and $11$ .

$8 x \geq 32$

Divide each term in $8 x \geq 32$ by $8$ and simplify.

Divide each term in $8 x \geq 32$ by $8$ .

$\frac{8 x}{8} \geq \frac{32}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{32}{8}$

Divide $x$ by $1$ .

$x \geq \frac{32}{8}$

Simplify the right side.

Divide $32$ by $8$ .

$x \geq 4$

Find the union of the solutions.

$x \leq - 3$ or $x \geq 4$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3 or x \geq 4$

Interval Notation:

$(- \infty, - 3] \cup [4, \infty)$

Solve for x 4|1-2x|-7>=21

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