Algebra Examples

$| 2 x - 1 | > | 3 x + 5 |$

Step 1

Replace $>$ with $=$ in $| 2 x - 1 | > | 3 x + 5 |$ .

$| 2 x - 1 | = | 3 x + 5 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$2 x - 1 = 3 x + 5$

$2 x - 1 = - (3 x + 5)$

$- (2 x - 1) = 3 x + 5$

$- (2 x - 1) = - (3 x + 5)$

Step 3

After simplifying, there are only two unique equations to be solved.

$2 x - 1 = 3 x + 5$

$2 x - 1 = - (3 x + 5)$

Step 4

Solve $2 x - 1 = 3 x + 5$ for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $3 x$ from both sides of the equation.

$2 x - 1 - 3 x = 5$

Step 4.1.2

Subtract $3 x$ from $2 x$ .

$- x - 1 = 5$

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

$- x = 5 + 1$

Step 4.2.2

Add $5$ and $1$ .

$- x = 6$

Step 4.3

Divide each term in $- x = 6$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- x = 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{6}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{6}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{6}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $6$ by $- 1$ .

$x = - 6$

Step 5

Solve $2 x - 1 = - (3 x + 5)$ for $x$ .

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Step 5.1

Simplify $- (3 x + 5)$ .

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Step 5.1.1

Rewrite.

$2 x - 1 = 0 + 0 - (3 x + 5)$

Step 5.1.2

Simplify by adding zeros.

$2 x - 1 = - (3 x + 5)$

Step 5.1.3

Apply the distributive property.

$2 x - 1 = - (3 x) - 1 \cdot 5$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $3$ by $- 1$ .

$2 x - 1 = - 3 x - 1 \cdot 5$

Step 5.1.4.2

Multiply $- 1$ by $5$ .

$2 x - 1 = - 3 x - 5$

Step 5.2

Move all terms containing $x$ to the left side of the equation.

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Step 5.2.1

Add $3 x$ to both sides of the equation.

$2 x - 1 + 3 x = - 5$

Step 5.2.2

Add $2 x$ and $3 x$ .

$5 x - 1 = - 5$

Step 5.3

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

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Step 5.3.1

Add $1$ to both sides of the equation.

$5 x = - 5 + 1$

Step 5.3.2

Add $- 5$ and $1$ .

$5 x = - 4$

Step 5.4

Divide each term in $5 x = - 4$ by $5$ and simplify.

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Step 5.4.1

Divide each term in $5 x = - 4$ by $5$ .

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $5$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{5}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

Step 6

List all of the solutions.

$x = - 6, - \frac{4}{5}$

Step 7

Use each root to create test intervals.

$x < - 6$

$- 6 < x < - \frac{4}{5}$

$x > - \frac{4}{5}$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.1

Test a value on the interval $x < - 6$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

Step 8.1.2

Replace $x$ with $- 8$ in the original inequality.

$| 2 (- 8) - 1 | > | 3 (- 8) + 5 |$

Step 8.1.3

The left side $17$ is not greater than the right side $19$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $- 6 < x < - \frac{4}{5}$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $- 6 < x < - \frac{4}{5}$ and see if this value makes the original inequality true.

$x = - 3$

Step 8.2.2

Replace $x$ with $- 3$ in the original inequality.

$| 2 (- 3) - 1 | > | 3 (- 3) + 5 |$

Step 8.2.3

The left side $7$ is greater than the right side $4$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $x > - \frac{4}{5}$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > - \frac{4}{5}$ and see if this value makes the original inequality true.

$x = 0$

Step 8.3.2

Replace $x$ with $0$ in the original inequality.

$| 2 (0) - 1 | > | 3 (0) + 5 |$

Step 8.3.3

The left side $1$ is not greater than the right side $5$ , which means that the given statement is false.

False

Step 8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 6$ False

$- 6 < x < - \frac{4}{5}$ True

$x > - \frac{4}{5}$ False

$x < - 6$ False

$- 6 < x < - \frac{4}{5}$ True

$x > - \frac{4}{5}$ False

Step 9

The solution consists of all of the true intervals.

$- 6 < x < - \frac{4}{5}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < - \frac{4}{5}$

Interval Notation:

$(- 6, - \frac{4}{5})$

Step 11