Precalculus Examples

$4 + | x - 5 | > \frac{5}{8}$

Step 1

Write $4 + | x - 5 | > \frac{5}{8}$ 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.

$x - 5 \geq 0$

Step 1.2

Add $5$ to both sides of the inequality.

$x \geq 5$

Step 1.3

In the piece where $x - 5$ is non-negative, remove the absolute value.

$4 + x - 5 > \frac{5}{8}$

Step 1.4

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

$x - 5 < 0$

Step 1.5

Add $5$ to both sides of the inequality.

$x < 5$

Step 1.6

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$4 - (x - 5) > \frac{5}{8}$

Step 1.7

Write as a piecewise.

${\begin{cases} 4 + x - 5 > \frac{5}{8} & x \geq 5 \\ 4 - (x - 5) > \frac{5}{8} & x < 5 \end{cases}$

Step 1.8

Subtract $5$ from $4$ .

${\begin{cases} x - 1 > \frac{5}{8} & x \geq 5 \\ 4 - (x - 5) > \frac{5}{8} & x < 5 \end{cases}$

Step 1.9

Simplify $4 - (x - 5) > \frac{5}{8}$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} x - 1 > \frac{5}{8} & x \geq 5 \\ 4 - x - - 5 > \frac{5}{8} & x < 5 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $- 5$ .

${\begin{cases} x - 1 > \frac{5}{8} & x \geq 5 \\ 4 - x + 5 > \frac{5}{8} & x < 5 \end{cases}$

Step 1.9.2

Add $4$ and $5$ .

${\begin{cases} x - 1 > \frac{5}{8} & x \geq 5 \\ - x + 9 > \frac{5}{8} & x < 5 \end{cases}$

Step 2

Solve $x - 1 > \frac{5}{8}$ when $x \geq 5$ .

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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.

$x > \frac{5}{8} + 1$

Step 2.1.2

Write $1$ as a fraction with a common denominator.

$x > \frac{5}{8} + \frac{8}{8}$

Step 2.1.3

Combine the numerators over the common denominator.

$x > \frac{5 + 8}{8}$

Step 2.1.4

Add $5$ and $8$ .

$x > \frac{13}{8}$

Step 2.2

Find the intersection of $x > \frac{13}{8}$ and $x \geq 5$ .

$x \geq 5$

Step 3

Solve $- x + 9 > \frac{5}{8}$ when $x < 5$ .

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Step 3.1

Solve $- x + 9 > \frac{5}{8}$ 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

Subtract $9$ from both sides of the inequality.

$- x > \frac{5}{8} - 9$

Step 3.1.1.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- x > \frac{5}{8} - 9 \cdot \frac{8}{8}$

Step 3.1.1.3

Combine $- 9$ and $\frac{8}{8}$ .

$- x > \frac{5}{8} + \frac{- 9 \cdot 8}{8}$

Step 3.1.1.4

Combine the numerators over the common denominator.

$- x > \frac{5 - 9 \cdot 8}{8}$

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

Multiply $- 9$ by $8$ .

$- x > \frac{5 - 72}{8}$

Step 3.1.1.5.2

Subtract $72$ from $5$ .

$- x > \frac{- 67}{8}$

Step 3.1.1.6

Move the negative in front of the fraction.

$- x > - \frac{67}{8}$

Step 3.1.2

Divide each term in $- x > - \frac{67}{8}$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- x > - \frac{67}{8}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- \frac{67}{8}}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- \frac{67}{8}}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x < \frac{- \frac{67}{8}}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Dividing two negative values results in a positive value.

$x < \frac{\frac{67}{8}}{1}$

Step 3.1.2.3.2

Divide $\frac{67}{8}$ by $1$ .

$x < \frac{67}{8}$

Step 3.2

Find the intersection of $x < \frac{67}{8}$ and $x < 5$ .

$x < 5$

Step 4

Find the union of the solutions.

All real numbers

Step 5

Convert the inequality to interval notation.

$(- \infty, \infty)$

Step 6