Algebra Examples

$2 | x + 2 | - 3 \geq 0$

Step 1

Write $2 | x + 2 | - 3 \geq 0$ as a piecewise.

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Step 1.1

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

$x + 2 \geq 0$

Step 1.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 1.3

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

$2 (x + 2) - 3 \geq 0$

Step 1.4

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

$x + 2 < 0$

Step 1.5

Subtract $2$ from both sides of the inequality.

$x < - 2$

Step 1.6

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

$2 (- (x + 2)) - 3 \geq 0$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 (x + 2) - 3 \geq 0 & x \geq - 2 \\ 2 (- (x + 2)) - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.8

Simplify $2 (x + 2) - 3 \geq 0$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 2 x + 2 \cdot 2 - 3 \geq 0 & x \geq - 2 \\ 2 (- (x + 2)) - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.8.1.2

Multiply $2$ by $2$ .

${\begin{cases} 2 x + 4 - 3 \geq 0 & x \geq - 2 \\ 2 (- (x + 2)) - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.8.2

Subtract $3$ from $4$ .

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ 2 (- (x + 2)) - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.9

Simplify $2 (- (x + 2)) - 3 \geq 0$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ 2 (- x - 1 \cdot 2) - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $2$ .

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ 2 (- x - 2) - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.9.1.3

Apply the distributive property.

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ 2 (- x) + 2 \cdot - 2 - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $2$ .

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ - 2 x + 2 \cdot - 2 - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.9.1.5

Multiply $2$ by $- 2$ .

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ - 2 x - 4 - 3 \geq 0 & x < - 2 \end{cases}$

Step 1.9.2

Subtract $3$ from $- 4$ .

${\begin{cases} 2 x + 1 \geq 0 & x \geq - 2 \\ - 2 x - 7 \geq 0 & x < - 2 \end{cases}$

Step 2

Solve $2 x + 1 \geq 0$ for $x$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$2 x \geq - 1$

Step 2.2

Divide each term in $2 x \geq - 1$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq - 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 1}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 1}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 1}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{1}{2}$

Step 3

Solve $- 2 x - 7 \geq 0$ for $x$ .

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Step 3.1

Add $7$ to both sides of the inequality.

$- 2 x \geq 7$

Step 3.2

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

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Step 3.2.1

Divide each term in $- 2 x \geq 7$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

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

Step 4

Find the union of the solutions.

$x \leq - \frac{7}{2}$ or $x \geq - \frac{1}{2}$

Step 5