Pre-Algebra Examples

$| 4 x + 3 | < 1$

Step 1

Write $| 4 x + 3 | < 1$ 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.

$4 x + 3 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $3$ from both sides of the inequality.

$4 x \geq - 3$

Step 1.2.2

Divide each term in $4 x \geq - 3$ by $4$ and simplify.

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Step 1.2.2.1

Divide each term in $4 x \geq - 3$ by $4$ .

$\frac{4 x}{4} \geq \frac{- 3}{4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{- 3}{4}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 3}{4}$

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.

$x \geq - \frac{3}{4}$

Step 1.3

In the piece where $4 x + 3$ is non-negative, remove the absolute value.

$4 x + 3 < 1$

Step 1.4

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

$4 x + 3 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $3$ from both sides of the inequality.

$4 x < - 3$

Step 1.5.2

Divide each term in $4 x < - 3$ by $4$ and simplify.

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Step 1.5.2.1

Divide each term in $4 x < - 3$ by $4$ .

$\frac{4 x}{4} < \frac{- 3}{4}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $4$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{- 3}{4}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 3}{4}$

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.

$x < - \frac{3}{4}$

Step 1.6

In the piece where $4 x + 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x + 3) < 1$

Step 1.7

Write as a piecewise.

${\begin{cases} 4 x + 3 < 1 & x \geq - \frac{3}{4} \\ - (4 x + 3) < 1 & x < - \frac{3}{4} \end{cases}$

Step 1.8

Simplify $- (4 x + 3) < 1$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} 4 x + 3 < 1 & x \geq - \frac{3}{4} \\ - (4 x) - 1 \cdot 3 < 1 & x < - \frac{3}{4} \end{cases}$

Step 1.8.2

Multiply $4$ by $- 1$ .

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

Step 1.8.3

Multiply $- 1$ by $3$ .

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

Step 2

Solve $4 x + 3 < 1$ when $x \geq - \frac{3}{4}$ .

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Step 2.1

Solve $4 x + 3 < 1$ 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 $3$ from both sides of the inequality.

$4 x < 1 - 3$

Step 2.1.1.2

Subtract $3$ from $1$ .

$4 x < - 2$

Step 2.1.2

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

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Step 2.1.2.1

Divide each term in $4 x < - 2$ by $4$ .

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

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

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

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

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

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Step 2.1.2.3.1.1

Factor $2$ out of $- 2$ .

$x < \frac{2 (- 1)}{4}$

Step 2.1.2.3.1.2

Cancel the common factors.

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Step 2.1.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 2.1.2.3.1.2.2

Cancel the common factor.

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

Step 2.1.2.3.1.2.3

Rewrite the expression.

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

Step 2.1.2.3.2

Move the negative in front of the fraction.

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

Step 2.2

Find the intersection of $x < - \frac{1}{2}$ and $x \geq - \frac{3}{4}$ .

$- \frac{3}{4} \leq x < - \frac{1}{2}$

Step 3

Solve $- 4 x - 3 < 1$ when $x < - \frac{3}{4}$ .

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Step 3.1

Solve $- 4 x - 3 < 1$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $3$ to both sides of the inequality.

$- 4 x < 1 + 3$

Step 3.1.1.2

Add $1$ and $3$ .

$- 4 x < 4$

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

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

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $4$ by $- 4$ .

$x > - 1$

Step 3.2

Find the intersection of $x > - 1$ and $x < - \frac{3}{4}$ .

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

Step 4

Find the union of the solutions.

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

Step 5