Precalculus Examples

$| 2 x + 1 | + 1 \leq 4$

Step 1

Write $| 2 x + 1 | + 1 \leq 4$ 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.

$2 x + 1 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $1$ from both sides of the inequality.

$2 x \geq - 1$

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.3

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

$2 x + 1 + 1 \leq 4$

Step 1.4

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

$2 x + 1 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $1$ from both sides of the inequality.

$2 x < - 1$

Step 1.5.2

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

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Step 1.5.2.1

Divide each term in $2 x < - 1$ by $2$ .

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Move the negative in front of the fraction.

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

Step 1.6

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

$- (2 x + 1) + 1 \leq 4$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 x + 1 + 1 \leq 4 & x \geq - \frac{1}{2} \\ - (2 x + 1) + 1 \leq 4 & x < - \frac{1}{2} \end{cases}$

Step 1.8

Add $1$ and $1$ .

${\begin{cases} 2 x + 2 \leq 4 & x \geq - \frac{1}{2} \\ - (2 x + 1) + 1 \leq 4 & x < - \frac{1}{2} \end{cases}$

Step 1.9

Simplify $- (2 x + 1) + 1 \leq 4$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 2 x + 2 \leq 4 & x \geq - \frac{1}{2} \\ - (2 x) - 1 \cdot 1 + 1 \leq 4 & x < - \frac{1}{2} \end{cases}$

Step 1.9.1.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 2 \leq 4 & x \geq - \frac{1}{2} \\ - 2 x - 1 \cdot 1 + 1 \leq 4 & x < - \frac{1}{2} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $1$ .

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

Step 1.9.2

Combine the opposite terms in $- 2 x - 1 + 1$ .

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Step 1.9.2.1

Add $- 1$ and $1$ .

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

Step 1.9.2.2

Add $- 2 x$ and $0$ .

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

Step 2

Solve $2 x + 2 \leq 4$ when $x \geq - \frac{1}{2}$ .

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Step 2.1

Solve $2 x + 2 \leq 4$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $2$ from both sides of the inequality.

$2 x \leq 4 - 2$

Step 2.1.1.2

Subtract $2$ from $4$ .

$2 x \leq 2$

Step 2.1.2

Divide each term in $2 x \leq 2$ by $2$ and simplify.

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Step 2.1.2.1

Divide each term in $2 x \leq 2$ by $2$ .

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

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

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

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $2$ by $2$ .

$x \leq 1$

Step 2.2

Find the intersection of $x \leq 1$ and $x \geq - \frac{1}{2}$ .

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

Step 3

Solve $- 2 x \leq 4$ when $x < - \frac{1}{2}$ .

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Step 3.1

Divide each term in $- 2 x \leq 4$ by $- 2$ and simplify.

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Step 3.1.1

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

$\frac{- 2 x}{- 2} \geq \frac{4}{- 2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{4}{- 2}$

Step 3.1.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{4}{- 2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $4$ by $- 2$ .

$x \geq - 2$

Step 3.2

Find the intersection of $x \geq - 2$ and $x < - \frac{1}{2}$ .

$- 2 \leq x < - \frac{1}{2}$

Step 4

Find the union of the solutions.

$- 2 \leq x \leq 1$

Step 5

Convert the inequality to interval notation.

$[- 2, 1]$

Step 6