Solve the Absolute Value Inequality for k |1-3/4k|>=7

$| 1 - \frac{3}{4} k | \geq 7$

Step 1

Write $| 1 - \frac{3}{4} k | \geq 7$ as a piecewise.

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

$1 - \frac{3}{4} k \geq 0$

Solve the inequality.

Simplify each term.

Combine $k$ and $\frac{3}{4}$ .

$1 - \frac{k \cdot 3}{4} \geq 0$

Move $3$ to the left of $k$ .

$1 - \frac{3 k}{4} \geq 0$

Subtract $1$ from both sides of the inequality.

$- \frac{3 k}{4} \geq - 1$

Divide each term in $- \frac{3 k}{4} \geq - 1$ by $- 1$ and simplify.

Divide each term in $- \frac{3 k}{4} \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 k}{4}}{- 1} \leq \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{3 k}{4}}{1} \leq \frac{- 1}{- 1}$

Divide $\frac{3 k}{4}$ by $1$ .

$\frac{3 k}{4} \leq \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$\frac{3 k}{4} \leq 1$

Multiply both sides by $4$ .

$\frac{3 k}{4} \cdot 4 \leq 1 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 k}{4} \cdot 4 \leq 1 \cdot 4$

Rewrite the expression.

$3 k \leq 1 \cdot 4$

Simplify the right side.

Multiply $4$ by $1$ .

$3 k \leq 4$

Divide each term in $3 k \leq 4$ by $3$ and simplify.

Divide each term in $3 k \leq 4$ by $3$ .

$\frac{3 k}{3} \leq \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 k}{3} \leq \frac{4}{3}$

Divide $k$ by $1$ .

$k \leq \frac{4}{3}$

In the piece where $1 - \frac{3}{4} k$ is non-negative, remove the absolute value.

$1 - \frac{3}{4} k \geq 7$

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

$1 - \frac{3}{4} k < 0$

Solve the inequality.

Simplify each term.

Combine $k$ and $\frac{3}{4}$ .

$1 - \frac{k \cdot 3}{4} < 0$

Move $3$ to the left of $k$ .

$1 - \frac{3 k}{4} < 0$

Subtract $1$ from both sides of the inequality.

$- \frac{3 k}{4} < - 1$

Divide each term in $- \frac{3 k}{4} < - 1$ by $- 1$ and simplify.

Divide each term in $- \frac{3 k}{4} < - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 k}{4}}{- 1} > \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{3 k}{4}}{1} > \frac{- 1}{- 1}$

Divide $\frac{3 k}{4}$ by $1$ .

$\frac{3 k}{4} > \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$\frac{3 k}{4} > 1$

Multiply both sides by $4$ .

$\frac{3 k}{4} \cdot 4 > 1 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 k}{4} \cdot 4 > 1 \cdot 4$

Rewrite the expression.

$3 k > 1 \cdot 4$

Simplify the right side.

Multiply $4$ by $1$ .

$3 k > 4$

Divide each term in $3 k > 4$ by $3$ and simplify.

Divide each term in $3 k > 4$ by $3$ .

$\frac{3 k}{3} > \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 k}{3} > \frac{4}{3}$

Divide $k$ by $1$ .

$k > \frac{4}{3}$

In the piece where $1 - \frac{3}{4} k$ is negative, remove the absolute value and multiply by $- 1$ .

$- (1 - \frac{3}{4} k) \geq 7$

Write as a piecewise.

${\begin{cases} 1 - \frac{3}{4} k \geq 7 & k \leq \frac{4}{3} \\ - (1 - \frac{3}{4} k) \geq 7 & k > \frac{4}{3} \end{cases}$

Simplify each term.

Combine $k$ and $\frac{3}{4}$ .

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

Move $3$ to the left of $k$ .

${\begin{cases} 1 - \frac{3 k}{4} \geq 7 & k \leq \frac{4}{3} \\ - (1 - \frac{3}{4} k) \geq 7 & k > \frac{4}{3} \end{cases}$

Simplify $- (1 - \frac{3}{4} k) \geq 7$ .

Simplify each term.

Combine $k$ and $\frac{3}{4}$ .

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

Move $3$ to the left of $k$ .

${\begin{cases} 1 - \frac{3 k}{4} \geq 7 & k \leq \frac{4}{3} \\ - (1 - \frac{3 k}{4}) \geq 7 & k > \frac{4}{3} \end{cases}$

Apply the distributive property.

${\begin{cases} 1 - \frac{3 k}{4} \geq 7 & k \leq \frac{4}{3} \\ - 1 \cdot 1 - - \frac{3 k}{4} \geq 7 & k > \frac{4}{3} \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} 1 - \frac{3 k}{4} \geq 7 & k \leq \frac{4}{3} \\ - 1 - - \frac{3 k}{4} \geq 7 & k > \frac{4}{3} \end{cases}$

Multiply $- - \frac{3 k}{4}$ .

Multiply $- 1$ by $- 1$ .

${\begin{cases} 1 - \frac{3 k}{4} \geq 7 & k \leq \frac{4}{3} \\ - 1 + 1 \frac{3 k}{4} \geq 7 & k > \frac{4}{3} \end{cases}$

Multiply $\frac{3 k}{4}$ by $1$ .

${\begin{cases} 1 - \frac{3 k}{4} \geq 7 & k \leq \frac{4}{3} \\ - 1 + \frac{3 k}{4} \geq 7 & k > \frac{4}{3} \end{cases}$

Step 2

Solve $1 - \frac{3 k}{4} \geq 7$ for $k$ .

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

Subtract $1$ from both sides of the inequality.

$- \frac{3 k}{4} \geq 7 - 1$

Subtract $1$ from $7$ .

$- \frac{3 k}{4} \geq 6$

Divide each term in $- \frac{3 k}{4} \geq 6$ by $- 1$ and simplify.

Divide each term in $- \frac{3 k}{4} \geq 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 k}{4}}{- 1} \leq \frac{6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{3 k}{4}}{1} \leq \frac{6}{- 1}$

Divide $\frac{3 k}{4}$ by $1$ .

$\frac{3 k}{4} \leq \frac{6}{- 1}$

Simplify the right side.

Divide $6$ by $- 1$ .

$\frac{3 k}{4} \leq - 6$

Multiply both sides by $4$ .

$\frac{3 k}{4} \cdot 4 \leq - 6 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 k}{4} \cdot 4 \leq - 6 \cdot 4$

Rewrite the expression.

$3 k \leq - 6 \cdot 4$

Simplify the right side.

Multiply $- 6$ by $4$ .

$3 k \leq - 24$

Divide each term in $3 k \leq - 24$ by $3$ and simplify.

Divide each term in $3 k \leq - 24$ by $3$ .

$\frac{3 k}{3} \leq \frac{- 24}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 k}{3} \leq \frac{- 24}{3}$

Divide $k$ by $1$ .

$k \leq \frac{- 24}{3}$

Simplify the right side.

Divide $- 24$ by $3$ .

$k \leq - 8$

Step 3

Solve $- 1 + \frac{3 k}{4} \geq 7$ for $k$ .

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

Add $1$ to both sides of the inequality.

$\frac{3 k}{4} \geq 7 + 1$

Add $7$ and $1$ .

$\frac{3 k}{4} \geq 8$

Multiply both sides by $4$ .

$\frac{3 k}{4} \cdot 4 \geq 8 \cdot 4$

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 k}{4} \cdot 4 \geq 8 \cdot 4$

Rewrite the expression.

$3 k \geq 8 \cdot 4$

Simplify the right side.

Multiply $8$ by $4$ .

$3 k \geq 32$

Divide each term in $3 k \geq 32$ by $3$ and simplify.

Divide each term in $3 k \geq 32$ by $3$ .

$\frac{3 k}{3} \geq \frac{32}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 k}{3} \geq \frac{32}{3}$

Divide $k$ by $1$ .

$k \geq \frac{32}{3}$

Step 4

Find the union of the solutions.

$k \leq - 8$ or $k \geq \frac{32}{3}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$k \leq - 8 or k \geq \frac{32}{3}$

Interval Notation:

$(- \infty, - 8] \cup [\frac{32}{3}, \infty)$

Step 6

Solve the Absolute Value Inequality for k |1-3/4k|>=7

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