Solve the Absolute Value Inequality for x |7x+4|>=74

$| 7 x + 4 | \geq 74$

Write $| 7 x + 4 | \geq 74$ as a piecewise.

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

$7 x + 4 \geq 0$

Solve the inequality.

Subtract $4$ from both sides of the inequality.

$7 x \geq - 4$

Divide each term in $7 x \geq - 4$ by $7$ and simplify.

Divide each term in $7 x \geq - 4$ by $7$ .

$\frac{7 x}{7} \geq \frac{- 4}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{- 4}{7}$

Divide $x$ by $1$ .

$x \geq \frac{- 4}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{4}{7}$

In the piece where $7 x + 4$ is non-negative, remove the absolute value.

$7 x + 4 \geq 74$

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

$7 x + 4 < 0$

Solve the inequality.

Subtract $4$ from both sides of the inequality.

$7 x < - 4$

Divide each term in $7 x < - 4$ by $7$ and simplify.

Divide each term in $7 x < - 4$ by $7$ .

$\frac{7 x}{7} < \frac{- 4}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 4}{7}$

Divide $x$ by $1$ .

$x < \frac{- 4}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x < - \frac{4}{7}$

In the piece where $7 x + 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (7 x + 4) \geq 74$

Write as a piecewise.

${\begin{cases} 7 x + 4 \geq 74 & x \geq - \frac{4}{7} \\ - (7 x + 4) \geq 74 & x < - \frac{4}{7} \end{cases}$

Simplify $- (7 x + 4) \geq 74$ .

Apply the distributive property.

${\begin{cases} 7 x + 4 \geq 74 & x \geq - \frac{4}{7} \\ - (7 x) - 1 \cdot 4 \geq 74 & x < - \frac{4}{7} \end{cases}$

Multiply $7$ by $- 1$ .

${\begin{cases} 7 x + 4 \geq 74 & x \geq - \frac{4}{7} \\ - 7 x - 1 \cdot 4 \geq 74 & x < - \frac{4}{7} \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} 7 x + 4 \geq 74 & x \geq - \frac{4}{7} \\ - 7 x - 4 \geq 74 & x < - \frac{4}{7} \end{cases}$

Solve $7 x + 4 \geq 74$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$7 x \geq 74 - 4$

Subtract $4$ from $74$ .

$7 x \geq 70$

Divide each term in $7 x \geq 70$ by $7$ and simplify.

Divide each term in $7 x \geq 70$ by $7$ .

$\frac{7 x}{7} \geq \frac{70}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{70}{7}$

Divide $x$ by $1$ .

$x \geq \frac{70}{7}$

Simplify the right side.

Divide $70$ by $7$ .

$x \geq 10$

Solve $- 7 x - 4 \geq 74$ for $x$ .

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

Add $4$ to both sides of the inequality.

$- 7 x \geq 74 + 4$

Add $74$ and $4$ .

$- 7 x \geq 78$

Divide each term in $- 7 x \geq 78$ by $- 7$ and simplify.

Divide each term in $- 7 x \geq 78$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{78}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{78}{- 7}$

Divide $x$ by $1$ .

$x \leq \frac{78}{- 7}$

Simplify the right side.

Move the negative in front of the fraction.

$x \leq - \frac{78}{7}$

Find the union of the solutions.

$x \leq - \frac{78}{7}$ or $x \geq 10$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{78}{7} or x \geq 10$

Interval Notation:

$(- \infty, - \frac{78}{7}] \cup [10, \infty)$

Solve the Absolute Value Inequality for x |7x+4|>=74

Related Questions

Support LCMGCF.com by disabling your adblocker.