Solve the Inequality for x |-2x+3|<=7

$| - 2 x + 3 | \leq 7$

Write $| - 2 x + 3 | \leq 7$ as a piecewise.

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

$- 2 x + 3 \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 2 x \geq - 3$

Divide each term in $- 2 x \geq - 3$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 3}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{3}{2}$

In the piece where $- 2 x + 3$ is non-negative, remove the absolute value.

$- 2 x + 3 \leq 7$

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

$- 2 x + 3 < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 2 x < - 3$

Divide each term in $- 2 x < - 3$ by $- 2$ and simplify.

Divide each term in $- 2 x < - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{- 3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{- 3}{- 2}$

Divide $x$ by $1$ .

$x > \frac{- 3}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{3}{2}$

In the piece where $- 2 x + 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 2 x + 3) \leq 7$

Write as a piecewise.

${\begin{cases} - 2 x + 3 \leq 7 & x \leq \frac{3}{2} \\ - (- 2 x + 3) \leq 7 & x > \frac{3}{2} \end{cases}$

Simplify $- (- 2 x + 3) \leq 7$ .

Apply the distributive property.

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

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 2 x + 3 \leq 7 & x \leq \frac{3}{2} \\ 2 x - 1 \cdot 3 \leq 7 & x > \frac{3}{2} \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} - 2 x + 3 \leq 7 & x \leq \frac{3}{2} \\ 2 x - 3 \leq 7 & x > \frac{3}{2} \end{cases}$

Solve $- 2 x + 3 \leq 7$ when $x \leq \frac{3}{2}$ .

Solve $- 2 x + 3 \leq 7$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

$- 2 x \leq 7 - 3$

Subtract $3$ from $7$ .

$- 2 x \leq 4$

Divide each term in $- 2 x \leq 4$ by $- 2$ and simplify.

Divide each term in $- 2 x \leq 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{4}{- 2}$

Divide $x$ by $1$ .

$x \geq \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$x \geq - 2$

Find the intersection of $x \geq - 2$ and $x \leq \frac{3}{2}$ .

$- 2 \leq x \leq \frac{3}{2}$

Solve $2 x - 3 \leq 7$ when $x > \frac{3}{2}$ .

Solve $2 x - 3 \leq 7$ for $x$ .

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

Add $3$ to both sides of the inequality.

$2 x \leq 7 + 3$

Add $7$ and $3$ .

$2 x \leq 10$

Divide each term in $2 x \leq 10$ by $2$ and simplify.

Divide each term in $2 x \leq 10$ by $2$ .

$\frac{2 x}{2} \leq \frac{10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{10}{2}$

Divide $x$ by $1$ .

$x \leq \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

$x \leq 5$

Find the intersection of $x \leq 5$ and $x > \frac{3}{2}$ .

$\frac{3}{2} < x \leq 5$

Find the union of the solutions.

$- 2 \leq x \leq 5$

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x \leq 5$

Interval Notation:

$[- 2, 5]$

Solve the Inequality for x |-2x+3|<=7

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