Pre-Algebra Examples

| 3 + 4 x | - 4 > 3

$| 3 + 4 x | - 4 > 3$

Step 1

Write

| 3 + 4 x | - 4 > 3

$| 3 + 4 x | - 4 > 3$ 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.

3 + 4 x \geq 0

$3 + 4 x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract

3

$3$ from both sides of the inequality.

4 x \geq - 3

$4 x \geq - 3$

Step 1.2.2

Divide each term in

4 x \geq - 3

$4 x \geq - 3$ by

4

$4$ and simplify.

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Step 1.2.2.1

Divide each term in

4 x \geq - 3

$4 x \geq - 3$ by

4

$4$ .

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

$4$ .

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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

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

$3 + 4 x - 4 > 3$

Step 1.4

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

$3 + 4 x < 0$

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

$3 + 4 x$ is negative, remove the absolute value and multiply by

$- 1$ .

$- (3 + 4 x) - 4 > 3$

Step 1.7

Write as a piecewise.

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

Step 1.8

Subtract

$4$ from

$3$ .

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

Step 1.9

Simplify

$- (3 + 4 x) - 4 > 3$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

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

Step 1.9.1.2

Multiply

$- 1$ by

$3$ .

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

Step 1.9.1.3

Multiply

$4$ by

$- 1$ .

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

Step 1.9.2

Subtract

$4$ from

$- 3$ .

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

Step 2

Solve

$4 x - 1 > 3$ for

$x$ .

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Step 2.1

Move all terms not containing

$x$ to the right side of the inequality.

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Step 2.1.1

Add

$1$ to both sides of the inequality.

$4 x > 3 + 1$

Step 2.1.2

Add

$3$ and

$1$ .

$4 x > 4$

Step 2.2

Divide each term in

$4 x > 4$ by

$4$ and simplify.

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Step 2.2.1

Divide each term in

$4 x > 4$ by

$4$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of

$4$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide

$4$ by

$4$ .

$x > 1$

Step 3

Solve

$- 4 x - 7 > 3$ for

$x$ .

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Step 3.1

Move all terms not containing

$x$ to the right side of the inequality.

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Step 3.1.1

Add

$7$ to both sides of the inequality.

$- 4 x > 3 + 7$

Step 3.1.2

Add

$3$ and

$7$ .

$- 4 x > 10$

Step 3.2

Divide each term in

$- 4 x > 10$ by

$- 4$ and simplify.

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Step 3.2.1

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.

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

$- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide

$x$ by

$1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of

$10$ and

$- 4$ .

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Step 3.2.3.1.1

Factor

$2$ out of

$10$ .

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

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor

$2$ out of

$- 4$ .

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

Step 3.2.3.1.2.2

Cancel the common factor.

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

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 3.2.3.2

Move the negative in front of the fraction.

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

Step 4

Find the union of the solutions.

$x < - \frac{5}{2}$ or

$x > 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{5}{2} or x > 1$

Interval Notation:

$(- \infty, - \frac{5}{2}) \cup (1, \infty)$

Step 6