Solve for x 3|2x+7|+7<4

$3 | 2 x + 7 | + 7 < 4$

Step 1

Write $3 | 2 x + 7 | + 7 < 4$ as a piecewise.

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

$2 x + 7 \geq 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$2 x \geq - 7$

Divide each term in $2 x \geq - 7$ by $2$ and simplify.

Divide each term in $2 x \geq - 7$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 7}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{7}{2}$

In the piece where $2 x + 7$ is non-negative, remove the absolute value.

$3 (2 x + 7) + 7 < 4$

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

$2 x + 7 < 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$2 x < - 7$

Divide each term in $2 x < - 7$ by $2$ and simplify.

Divide each term in $2 x < - 7$ by $2$ .

$\frac{2 x}{2} < \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 7}{2}$

Divide $x$ by $1$ .

$x < \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{7}{2}$

In the piece where $2 x + 7$ is negative, remove the absolute value and multiply by $- 1$ .

$3 (- (2 x + 7)) + 7 < 4$

Write as a piecewise.

${\begin{cases} 3 (2 x + 7) + 7 < 4 & x \geq - \frac{7}{2} \\ 3 (- (2 x + 7)) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Simplify $3 (2 x + 7) + 7 < 4$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 (2 x) + 3 \cdot 7 + 7 < 4 & x \geq - \frac{7}{2} \\ 3 (- (2 x + 7)) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Multiply $2$ by $3$ .

${\begin{cases} 6 x + 3 \cdot 7 + 7 < 4 & x \geq - \frac{7}{2} \\ 3 (- (2 x + 7)) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Multiply $3$ by $7$ .

${\begin{cases} 6 x + 21 + 7 < 4 & x \geq - \frac{7}{2} \\ 3 (- (2 x + 7)) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Add $21$ and $7$ .

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ 3 (- (2 x + 7)) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Simplify $3 (- (2 x + 7)) + 7 < 4$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ 3 (- (2 x) - 1 \cdot 7) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ 3 (- 2 x - 1 \cdot 7) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ 3 (- 2 x - 7) + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ 3 (- 2 x) + 3 \cdot - 7 + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Multiply $- 2$ by $3$ .

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ - 6 x + 3 \cdot - 7 + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Multiply $3$ by $- 7$ .

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ - 6 x - 21 + 7 < 4 & x < - \frac{7}{2} \end{cases}$

Add $- 21$ and $7$ .

${\begin{cases} 6 x + 28 < 4 & x \geq - \frac{7}{2} \\ - 6 x - 14 < 4 & x < - \frac{7}{2} \end{cases}$

Step 2

Solve $6 x + 28 < 4$ when $x \geq - \frac{7}{2}$ .

Solve $6 x + 28 < 4$ for $x$ .

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

Subtract $28$ from both sides of the inequality.

$6 x < 4 - 28$

Subtract $28$ from $4$ .

$6 x < - 24$

Divide each term in $6 x < - 24$ by $6$ and simplify.

Divide each term in $6 x < - 24$ by $6$ .

$\frac{6 x}{6} < \frac{- 24}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} < \frac{- 24}{6}$

Divide $x$ by $1$ .

$x < \frac{- 24}{6}$

Simplify the right side.

Divide $- 24$ by $6$ .

$x < - 4$

Find the intersection of $x < - 4$ and $x \geq - \frac{7}{2}$ .

No solution

Step 3

Solve $- 6 x - 14 < 4$ when $x < - \frac{7}{2}$ .

Solve $- 6 x - 14 < 4$ for $x$ .

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

Add $14$ to both sides of the inequality.

$- 6 x < 4 + 14$

Add $4$ and $14$ .

$- 6 x < 18$

Divide each term in $- 6 x < 18$ by $- 6$ and simplify.

Divide each term in $- 6 x < 18$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} > \frac{18}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} > \frac{18}{- 6}$

Divide $x$ by $1$ .

$x > \frac{18}{- 6}$

Simplify the right side.

Divide $18$ by $- 6$ .

$x > - 3$

Find the intersection of $x > - 3$ and $x < - \frac{7}{2}$ .

No solution

Step 4

Find the union of the solutions.

No solution

Solve for x 3|2x+7|+7<4

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