Basic Math Examples

$17 - | 7 x + 2 | \geq 1$

Step 1

Write $17 - | 7 x + 2 | \geq 1$ 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.

$7 x + 2 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $2$ from both sides of the inequality.

$7 x \geq - 2$

Step 1.2.2

Divide each term in $7 x \geq - 2$ by $7$ and simplify.

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Step 1.2.2.1

Divide each term in $7 x \geq - 2$ by $7$ .

$\frac{7 x}{7} \geq \frac{- 2}{7}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{- 2}{7}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 2}{7}$

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.

$x \geq - \frac{2}{7}$

Step 1.3

In the piece where $7 x + 2$ is non-negative, remove the absolute value.

$17 - (7 x + 2) \geq 1$

Step 1.4

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

$7 x + 2 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $2$ from both sides of the inequality.

$7 x < - 2$

Step 1.5.2

Divide each term in $7 x < - 2$ by $7$ and simplify.

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Step 1.5.2.1

Divide each term in $7 x < - 2$ by $7$ .

$\frac{7 x}{7} < \frac{- 2}{7}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $7$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 2}{7}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 2}{7}$

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.

$x < - \frac{2}{7}$

Step 1.6

In the piece where $7 x + 2$ is negative, remove the absolute value and multiply by $- 1$ .

$17 - - (7 x + 2) \geq 1$

Step 1.7

Write as a piecewise.

${\begin{cases} 17 - (7 x + 2) \geq 1 & x \geq - \frac{2}{7} \\ 17 - - (7 x + 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.8

Simplify $17 - (7 x + 2) \geq 1$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 17 - (7 x) - 1 \cdot 2 \geq 1 & x \geq - \frac{2}{7} \\ 17 - - (7 x + 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.8.1.2

Multiply $7$ by $- 1$ .

${\begin{cases} 17 - 7 x - 1 \cdot 2 \geq 1 & x \geq - \frac{2}{7} \\ 17 - - (7 x + 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.8.1.3

Multiply $- 1$ by $2$ .

${\begin{cases} 17 - 7 x - 2 \geq 1 & x \geq - \frac{2}{7} \\ 17 - - (7 x + 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.8.2

Subtract $2$ from $17$ .

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 - - (7 x + 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9

Simplify $17 - - (7 x + 2) \geq 1$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 - (- (7 x) - 1 \cdot 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9.1.2

Multiply $7$ by $- 1$ .

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 - (- 7 x - 1 \cdot 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $2$ .

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 - (- 7 x - 2) \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9.1.4

Apply the distributive property.

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 - (- 7 x) - - 2 \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9.1.5

Multiply $- 7$ by $- 1$ .

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 + 7 x - - 2 \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9.1.6

Multiply $- 1$ by $- 2$ .

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 17 + 7 x + 2 \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 1.9.2

Add $17$ and $2$ .

${\begin{cases} - 7 x + 15 \geq 1 & x \geq - \frac{2}{7} \\ 7 x + 19 \geq 1 & x < - \frac{2}{7} \end{cases}$

Step 2

Solve $- 7 x + 15 \geq 1$ when $x \geq - \frac{2}{7}$ .

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Step 2.1

Solve $- 7 x + 15 \geq 1$ 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 $15$ from both sides of the inequality.

$- 7 x \geq 1 - 15$

Step 2.1.1.2

Subtract $15$ from $1$ .

$- 7 x \geq - 14$

Step 2.1.2

Divide each term in $- 7 x \geq - 14$ by $- 7$ and simplify.

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Step 2.1.2.1

Divide each term in $- 7 x \geq - 14$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{- 14}{- 7}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $- 7$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{- 14}{- 7}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 14}{- 7}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $- 14$ by $- 7$ .

$x \leq 2$

Step 2.2

Find the intersection of $x \leq 2$ and $x \geq - \frac{2}{7}$ .

$- \frac{2}{7} \leq x \leq 2$

Step 3

Solve $7 x + 19 \geq 1$ when $x < - \frac{2}{7}$ .

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Step 3.1

Solve $7 x + 19 \geq 1$ 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

Subtract $19$ from both sides of the inequality.

$7 x \geq 1 - 19$

Step 3.1.1.2

Subtract $19$ from $1$ .

$7 x \geq - 18$

Step 3.1.2

Divide each term in $7 x \geq - 18$ by $7$ and simplify.

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Step 3.1.2.1

Divide each term in $7 x \geq - 18$ by $7$ .

$\frac{7 x}{7} \geq \frac{- 18}{7}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $7$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{- 18}{7}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 18}{7}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{18}{7}$

Step 3.2

Find the intersection of $x \geq - \frac{18}{7}$ and $x < - \frac{2}{7}$ .

$- \frac{18}{7} \leq x < - \frac{2}{7}$

Step 4

Find the union of the solutions.

$- \frac{18}{7} \leq x \leq 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{18}{7} \leq x \leq 2$

Interval Notation:

$[- \frac{18}{7}, 2]$

Step 6