Precalculus Examples

$| x - 8.2 | - 1.5 \leq 0$

Step 1

Write $| x - 8.2 | - 1.5 \leq 0$ 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.

$x - 8.2 \geq 0$

Step 1.2

Add $8.2$ to both sides of the inequality.

$x \geq 8.2$

Step 1.3

In the piece where $x - 8.2$ is non-negative, remove the absolute value.

$x - 8.2 - 1.5 \leq 0$

Step 1.4

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

$x - 8.2 < 0$

Step 1.5

Add $8.2$ to both sides of the inequality.

$x < 8.2$

Step 1.6

In the piece where $x - 8.2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 8.2) - 1.5 \leq 0$

Step 1.7

Write as a piecewise.

${\begin{cases} x - 8.2 - 1.5 \leq 0 & x \geq 8.2 \\ - (x - 8.2) - 1.5 \leq 0 & x < 8.2 \end{cases}$

Step 1.8

Subtract $1.5$ from $- 8.2$ .

${\begin{cases} x - 9.7 \leq 0 & x \geq 8.2 \\ - (x - 8.2) - 1.5 \leq 0 & x < 8.2 \end{cases}$

Step 1.9

Simplify $- (x - 8.2) - 1.5 \leq 0$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} x - 9.7 \leq 0 & x \geq 8.2 \\ - x - - 8.2 - 1.5 \leq 0 & x < 8.2 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $- 8.2$ .

${\begin{cases} x - 9.7 \leq 0 & x \geq 8.2 \\ - x + 8.2 - 1.5 \leq 0 & x < 8.2 \end{cases}$

Step 1.9.2

Subtract $1.5$ from $8.2$ .

${\begin{cases} x - 9.7 \leq 0 & x \geq 8.2 \\ - x + 6.7 \leq 0 & x < 8.2 \end{cases}$

Step 2

Solve $x - 9.7 \leq 0$ when $x \geq 8.2$ .

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Step 2.1

Add $9.7$ to both sides of the inequality.

$x \leq 9.7$

Step 2.2

Find the intersection of $x \leq 9.7$ and $x \geq 8.2$ .

$8.2 \leq x \leq 9.7$

Step 3

Solve $- x + 6.7 \leq 0$ when $x < 8.2$ .

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Step 3.1

Solve $- x + 6.7 \leq 0$ for $x$ .

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Step 3.1.1

Subtract $6.7$ from both sides of the inequality.

$- x \leq - 6.7$

Step 3.1.2

Divide each term in $- x \leq - 6.7$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- x \leq - 6.7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{- 6.7}{- 1}$

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.

$\frac{x}{1} \geq \frac{- 6.7}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x \geq \frac{- 6.7}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 6.7$ by $- 1$ .

$x \geq 6.7$

Step 3.2

Find the intersection of $x \geq 6.7$ and $x < 8.2$ .

$6.7 \leq x < 8.2$

Step 4

Find the union of the solutions.

$6.7 \leq x \leq 9.7$

Step 5

Convert the inequality to interval notation.

$[6.7, 9.7]$

Step 6