Algebra Examples

$8 + | 4 v - 7 | \geq 17$

Step 1

Write $8 + | 4 v - 7 | \geq 17$ as a piecewise.

Tap for more steps...

Step 1.1

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

$4 v - 7 \geq 0$

Step 1.2

Solve the inequality.

Tap for more steps...

Step 1.2.1

Add $7$ to both sides of the inequality.

$4 v \geq 7$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

Divide each term in $4 v \geq 7$ by $4$ .

$\frac{4 v}{4} \geq \frac{7}{4}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 v}{4} \geq \frac{7}{4}$

Step 1.2.2.2.1.2

Divide $v$ by $1$ .

$v \geq \frac{7}{4}$

Step 1.3

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

$8 + 4 v - 7 \geq 17$

Step 1.4

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

$4 v - 7 < 0$

Step 1.5

Solve the inequality.

Tap for more steps...

Step 1.5.1

Add $7$ to both sides of the inequality.

$4 v < 7$

Step 1.5.2

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

Tap for more steps...

Step 1.5.2.1

Divide each term in $4 v < 7$ by $4$ .

$\frac{4 v}{4} < \frac{7}{4}$

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 v}{4} < \frac{7}{4}$

Step 1.5.2.2.1.2

Divide $v$ by $1$ .

$v < \frac{7}{4}$

Step 1.6

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

$8 - (4 v - 7) \geq 17$

Step 1.7

Write as a piecewise.

${\begin{cases} 8 + 4 v - 7 \geq 17 & v \geq \frac{7}{4} \\ 8 - (4 v - 7) \geq 17 & v < \frac{7}{4} \end{cases}$

Step 1.8

Subtract $7$ from $8$ .

${\begin{cases} 4 v + 1 \geq 17 & v \geq \frac{7}{4} \\ 8 - (4 v - 7) \geq 17 & v < \frac{7}{4} \end{cases}$

Step 1.9

Simplify $8 - (4 v - 7) \geq 17$ .

Tap for more steps...

Step 1.9.1

Simplify each term.

Tap for more steps...

Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 4 v + 1 \geq 17 & v \geq \frac{7}{4} \\ 8 - (4 v) - - 7 \geq 17 & v < \frac{7}{4} \end{cases}$

Step 1.9.1.2

Multiply $4$ by $- 1$ .

${\begin{cases} 4 v + 1 \geq 17 & v \geq \frac{7}{4} \\ 8 - 4 v - - 7 \geq 17 & v < \frac{7}{4} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 7$ .

${\begin{cases} 4 v + 1 \geq 17 & v \geq \frac{7}{4} \\ 8 - 4 v + 7 \geq 17 & v < \frac{7}{4} \end{cases}$

Step 1.9.2

Add $8$ and $7$ .

${\begin{cases} 4 v + 1 \geq 17 & v \geq \frac{7}{4} \\ - 4 v + 15 \geq 17 & v < \frac{7}{4} \end{cases}$

Step 2

Solve $4 v + 1 \geq 17$ for $v$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Subtract $1$ from both sides of the inequality.

$4 v \geq 17 - 1$

Step 2.1.2

Subtract $1$ from $17$ .

$4 v \geq 16$

Step 2.2

Divide each term in $4 v \geq 16$ by $4$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $4 v \geq 16$ by $4$ .

$\frac{4 v}{4} \geq \frac{16}{4}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 v}{4} \geq \frac{16}{4}$

Step 2.2.2.1.2

Divide $v$ by $1$ .

$v \geq \frac{16}{4}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $16$ by $4$ .

$v \geq 4$

Step 3

Solve $- 4 v + 15 \geq 17$ for $v$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $15$ from both sides of the inequality.

$- 4 v \geq 17 - 15$

Step 3.1.2

Subtract $15$ from $17$ .

$- 4 v \geq 2$

Step 3.2

Divide each term in $- 4 v \geq 2$ by $- 4$ and simplify.

Tap for more steps...

Step 3.2.1

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

$\frac{- 4 v}{- 4} \leq \frac{2}{- 4}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 v}{- 4} \leq \frac{2}{- 4}$

Step 3.2.2.1.2

Divide $v$ by $1$ .

$v \leq \frac{2}{- 4}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Cancel the common factor of $2$ and $- 4$ .

Tap for more steps...

Step 3.2.3.1.1

Factor $2$ out of $2$ .

$v \leq \frac{2 (1)}{- 4}$

Step 3.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1.2.1

Factor $2$ out of $- 4$ .

$v \leq \frac{2 \cdot 1}{2 \cdot - 2}$

Step 3.2.3.1.2.2

Cancel the common factor.

$v \leq \frac{2 \cdot 1}{2 \cdot - 2}$

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 3.2.3.2

Move the negative in front of the fraction.

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

Step 4

Find the union of the solutions.

$v \leq - \frac{1}{2}$ or $v \geq 4$

Step 5