Convert to Interval Notation |10-4x|>=20

$| 10 - 4 x | \geq 20$

Step 1

Write $| 10 - 4 x | \geq 20$ as a piecewise.

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

$10 - 4 x \geq 0$

Solve the inequality.

Subtract $10$ from both sides of the inequality.

$- 4 x \geq - 10$

Divide each term in $- 4 x \geq - 10$ by $- 4$ and simplify.

Divide each term in $- 4 x \geq - 10$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 10}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 10}{- 4}$

Divide $x$ by $1$ .

$x \leq \frac{- 10}{- 4}$

Simplify the right side.

Cancel the common factor of $- 10$ and $- 4$ .

Factor $- 2$ out of $- 10$ .

$x \leq \frac{- 2 (5)}{- 4}$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

$x \leq \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Cancel the common factor.

$x \leq \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Rewrite the expression.

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

In the piece where $10 - 4 x$ is non-negative, remove the absolute value.

$10 - 4 x \geq 20$

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

$10 - 4 x < 0$

Solve the inequality.

Subtract $10$ from both sides of the inequality.

$- 4 x < - 10$

Divide each term in $- 4 x < - 10$ by $- 4$ and simplify.

Divide each term in $- 4 x < - 10$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{- 10}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{- 10}{- 4}$

Divide $x$ by $1$ .

$x > \frac{- 10}{- 4}$

Simplify the right side.

Cancel the common factor of $- 10$ and $- 4$ .

Factor $- 2$ out of $- 10$ .

$x > \frac{- 2 (5)}{- 4}$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

$x > \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Cancel the common factor.

$x > \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Rewrite the expression.

$x > \frac{5}{2}$

In the piece where $10 - 4 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (10 - 4 x) \geq 20$

Write as a piecewise.

${\begin{cases} 10 - 4 x \geq 20 & x \leq \frac{5}{2} \\ - (10 - 4 x) \geq 20 & x > \frac{5}{2} \end{cases}$

Simplify $- (10 - 4 x) \geq 20$ .

Apply the distributive property.

${\begin{cases} 10 - 4 x \geq 20 & x \leq \frac{5}{2} \\ - 1 \cdot 10 - (- 4 x) \geq 20 & x > \frac{5}{2} \end{cases}$

Multiply $- 1$ by $10$ .

${\begin{cases} 10 - 4 x \geq 20 & x \leq \frac{5}{2} \\ - 10 - (- 4 x) \geq 20 & x > \frac{5}{2} \end{cases}$

Multiply $- 4$ by $- 1$ .

${\begin{cases} 10 - 4 x \geq 20 & x \leq \frac{5}{2} \\ - 10 + 4 x \geq 20 & x > \frac{5}{2} \end{cases}$

Step 2

Solve $10 - 4 x \geq 20$ for $x$ .

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

Subtract $10$ from both sides of the inequality.

$- 4 x \geq 20 - 10$

Subtract $10$ from $20$ .

$- 4 x \geq 10$

Divide each term in $- 4 x \geq 10$ by $- 4$ and simplify.

Divide each term in $- 4 x \geq 10$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{10}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{10}{- 4}$

Divide $x$ by $1$ .

$x \leq \frac{10}{- 4}$

Simplify the right side.

Cancel the common factor of $10$ and $- 4$ .

Factor $2$ out of $10$ .

$x \leq \frac{2 (5)}{- 4}$

Cancel the common factors.

Factor $2$ out of $- 4$ .

$x \leq \frac{2 \cdot 5}{2 \cdot - 2}$

Cancel the common factor.

$x \leq \frac{2 \cdot 5}{2 \cdot - 2}$

Rewrite the expression.

$x \leq \frac{5}{- 2}$

Move the negative in front of the fraction.

$x \leq - \frac{5}{2}$

Step 3

Solve $- 10 + 4 x \geq 20$ for $x$ .

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

Add $10$ to both sides of the inequality.

$4 x \geq 20 + 10$

Add $20$ and $10$ .

$4 x \geq 30$

Divide each term in $4 x \geq 30$ by $4$ and simplify.

Divide each term in $4 x \geq 30$ by $4$ .

$\frac{4 x}{4} \geq \frac{30}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{30}{4}$

Divide $x$ by $1$ .

$x \geq \frac{30}{4}$

Simplify the right side.

Cancel the common factor of $30$ and $4$ .

Factor $2$ out of $30$ .

$x \geq \frac{2 (15)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x \geq \frac{2 \cdot 15}{2 \cdot 2}$

Cancel the common factor.

$x \geq \frac{2 \cdot 15}{2 \cdot 2}$

Rewrite the expression.

$x \geq \frac{15}{2}$

Step 4

Find the union of the solutions.

$x \leq - \frac{5}{2}$ or $x \geq \frac{15}{2}$

Step 5

Convert the inequality to interval notation.

$(- \infty, - \frac{5}{2}] \cup [\frac{15}{2}, \infty)$

Step 6

Convert to Interval Notation |10-4x|>=20

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