Algebra Examples

$| \frac{3 - 7 x}{9} | \geq \frac{3}{5}$

Step 1

Write $| \frac{3 - 7 x}{9} | \geq \frac{3}{5}$ 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.

$\frac{3 - 7 x}{9} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $9$ .

$\frac{3 - 7 x}{9} \cdot 9 \geq 0 \cdot 9$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Simplify $\frac{3 - 7 x}{9} \cdot 9$ .

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Step 1.2.2.1.1.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1.1.1

Cancel the common factor.

$\frac{3 - 7 x}{9} \cdot 9 \geq 0 \cdot 9$

Step 1.2.2.1.1.1.2

Rewrite the expression.

$3 - 7 x \geq 0 \cdot 9$

Step 1.2.2.1.1.2

Reorder $3$ and $- 7 x$ .

$- 7 x + 3 \geq 0 \cdot 9$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $9$ .

$- 7 x + 3 \geq 0$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Subtract $3$ from both sides of the inequality.

$- 7 x \geq - 3$

Step 1.2.3.2

Divide each term in $- 7 x \geq - 3$ by $- 7$ and simplify.

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Step 1.2.3.2.1

Divide each term in $- 7 x \geq - 3$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{- 3}{- 7}$

Step 1.2.3.2.2

Simplify the left side.

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Step 1.2.3.2.2.1

Cancel the common factor of $- 7$ .

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Step 1.2.3.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{- 3}{- 7}$

Step 1.2.3.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 7}$

Step 1.2.3.2.3

Simplify the right side.

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Step 1.2.3.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{3}{7}$

Step 1.3

In the piece where $\frac{3 - 7 x}{9}$ is non-negative, remove the absolute value.

$\frac{3 - 7 x}{9} \geq \frac{3}{5}$

Step 1.4

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

$\frac{3 - 7 x}{9} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $9$ .

$\frac{3 - 7 x}{9} \cdot 9 < 0 \cdot 9$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Simplify the left side.

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Step 1.5.2.1.1

Simplify $\frac{3 - 7 x}{9} \cdot 9$ .

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Step 1.5.2.1.1.1

Cancel the common factor of $9$ .

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Step 1.5.2.1.1.1.1

Cancel the common factor.

$\frac{3 - 7 x}{9} \cdot 9 < 0 \cdot 9$

Step 1.5.2.1.1.1.2

Rewrite the expression.

$3 - 7 x < 0 \cdot 9$

Step 1.5.2.1.1.2

Reorder $3$ and $- 7 x$ .

$- 7 x + 3 < 0 \cdot 9$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $9$ .

$- 7 x + 3 < 0$

Step 1.5.3

Solve for $x$ .

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Step 1.5.3.1

Subtract $3$ from both sides of the inequality.

$- 7 x < - 3$

Step 1.5.3.2

Divide each term in $- 7 x < - 3$ by $- 7$ and simplify.

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Step 1.5.3.2.1

Divide each term in $- 7 x < - 3$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} > \frac{- 3}{- 7}$

Step 1.5.3.2.2

Simplify the left side.

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Step 1.5.3.2.2.1

Cancel the common factor of $- 7$ .

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Step 1.5.3.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} > \frac{- 3}{- 7}$

Step 1.5.3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 3}{- 7}$

Step 1.5.3.2.3

Simplify the right side.

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Step 1.5.3.2.3.1

Dividing two negative values results in a positive value.

$x > \frac{3}{7}$

Step 1.6

In the piece where $\frac{3 - 7 x}{9}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{3 - 7 x}{9} \geq \frac{3}{5}$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{3 - 7 x}{9} \geq \frac{3}{5} & x \leq \frac{3}{7} \\ - \frac{3 - 7 x}{9} \geq \frac{3}{5} & x > \frac{3}{7} \end{cases}$

Step 2

Solve $\frac{3 - 7 x}{9} \geq \frac{3}{5}$ for $x$ .

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Step 2.1

Multiply both sides by $9$ .

$\frac{3 - 7 x}{9} \cdot 9 \geq \frac{3}{5} \cdot 9$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{3 - 7 x}{9} \cdot 9$ .

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Step 2.2.1.1.1

Cancel the common factor of $9$ .

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Step 2.2.1.1.1.1

Cancel the common factor.

$\frac{3 - 7 x}{9} \cdot 9 \geq \frac{3}{5} \cdot 9$

Step 2.2.1.1.1.2

Rewrite the expression.

$3 - 7 x \geq \frac{3}{5} \cdot 9$

Step 2.2.1.1.2

Reorder $3$ and $- 7 x$ .

$- 7 x + 3 \geq \frac{3}{5} \cdot 9$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $\frac{3}{5} \cdot 9$ .

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Step 2.2.2.1.1

Combine $\frac{3}{5}$ and $9$ .

$- 7 x + 3 \geq \frac{3 \cdot 9}{5}$

Step 2.2.2.1.2

Multiply $3$ by $9$ .

$- 7 x + 3 \geq \frac{27}{5}$

Step 2.3

Solve for $x$ .

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Step 2.3.1

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

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Step 2.3.1.1

Subtract $3$ from both sides of the inequality.

$- 7 x \geq \frac{27}{5} - 3$

Step 2.3.1.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 7 x \geq \frac{27}{5} - 3 \cdot \frac{5}{5}$

Step 2.3.1.3

Combine $- 3$ and $\frac{5}{5}$ .

$- 7 x \geq \frac{27}{5} + \frac{- 3 \cdot 5}{5}$

Step 2.3.1.4

Combine the numerators over the common denominator.

$- 7 x \geq \frac{27 - 3 \cdot 5}{5}$

Step 2.3.1.5

Simplify the numerator.

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Step 2.3.1.5.1

Multiply $- 3$ by $5$ .

$- 7 x \geq \frac{27 - 15}{5}$

Step 2.3.1.5.2

Subtract $15$ from $27$ .

$- 7 x \geq \frac{12}{5}$

Step 2.3.2

Divide each term in $- 7 x \geq \frac{12}{5}$ by $- 7$ and simplify.

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Step 2.3.2.1

Divide each term in $- 7 x \geq \frac{12}{5}$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{\frac{12}{5}}{- 7}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 7$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{\frac{12}{5}}{- 7}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{\frac{12}{5}}{- 7}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{12}{5} \cdot \frac{1}{- 7}$

Step 2.3.2.3.2

Move the negative in front of the fraction.

$x \leq \frac{12}{5} (- \frac{1}{7})$

Step 2.3.2.3.3

Multiply $\frac{12}{5} (- \frac{1}{7})$ .

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Step 2.3.2.3.3.1

Multiply $\frac{12}{5}$ by $\frac{1}{7}$ .

$x \leq - \frac{12}{5 \cdot 7}$

Step 2.3.2.3.3.2

Multiply $5$ by $7$ .

$x \leq - \frac{12}{35}$

Step 3

Solve $- \frac{3 - 7 x}{9} \geq \frac{3}{5}$ for $x$ .

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Step 3.1

Divide each term in $- \frac{3 - 7 x}{9} \geq \frac{3}{5}$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \frac{3 - 7 x}{9} \geq \frac{3}{5}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 - 7 x}{9}}{- 1} \leq \frac{\frac{3}{5}}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{3 - 7 x}{9}}{1} \leq \frac{\frac{3}{5}}{- 1}$

Step 3.1.2.2

Divide $\frac{3 - 7 x}{9}$ by $1$ .

$\frac{3 - 7 x}{9} \leq \frac{\frac{3}{5}}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Move the negative one from the denominator of $\frac{\frac{3}{5}}{- 1}$ .

$\frac{3 - 7 x}{9} \leq - 1 \cdot \frac{3}{5}$

Step 3.1.3.2

Rewrite $- 1 \cdot \frac{3}{5}$ as $- \frac{3}{5}$ .

$\frac{3 - 7 x}{9} \leq - \frac{3}{5}$

Step 3.2

Multiply both sides by $9$ .

$\frac{3 - 7 x}{9} \cdot 9 \leq - \frac{3}{5} \cdot 9$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{3 - 7 x}{9} \cdot 9$ .

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Step 3.3.1.1.1

Cancel the common factor of $9$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{3 - 7 x}{9} \cdot 9 \leq - \frac{3}{5} \cdot 9$

Step 3.3.1.1.1.2

Rewrite the expression.

$3 - 7 x \leq - \frac{3}{5} \cdot 9$

Step 3.3.1.1.2

Reorder $3$ and $- 7 x$ .

$- 7 x + 3 \leq - \frac{3}{5} \cdot 9$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{3}{5} \cdot 9$ .

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Step 3.3.2.1.1

Multiply $- \frac{3}{5} \cdot 9$ .

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Step 3.3.2.1.1.1

Multiply $9$ by $- 1$ .

$- 7 x + 3 \leq - 9 (\frac{3}{5})$

Step 3.3.2.1.1.2

Combine $- 9$ and $\frac{3}{5}$ .

$- 7 x + 3 \leq \frac{- 9 \cdot 3}{5}$

Step 3.3.2.1.1.3

Multiply $- 9$ by $3$ .

$- 7 x + 3 \leq \frac{- 27}{5}$

Step 3.3.2.1.2

Move the negative in front of the fraction.

$- 7 x + 3 \leq - \frac{27}{5}$

Step 3.4

Solve for $x$ .

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Step 3.4.1

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

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Step 3.4.1.1

Subtract $3$ from both sides of the inequality.

$- 7 x \leq - \frac{27}{5} - 3$

Step 3.4.1.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 7 x \leq - \frac{27}{5} - 3 \cdot \frac{5}{5}$

Step 3.4.1.3

Combine $- 3$ and $\frac{5}{5}$ .

$- 7 x \leq - \frac{27}{5} + \frac{- 3 \cdot 5}{5}$

Step 3.4.1.4

Combine the numerators over the common denominator.

$- 7 x \leq \frac{- 27 - 3 \cdot 5}{5}$

Step 3.4.1.5

Simplify the numerator.

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Step 3.4.1.5.1

Multiply $- 3$ by $5$ .

$- 7 x \leq \frac{- 27 - 15}{5}$

Step 3.4.1.5.2

Subtract $15$ from $- 27$ .

$- 7 x \leq \frac{- 42}{5}$

Step 3.4.1.6

Move the negative in front of the fraction.

$- 7 x \leq - \frac{42}{5}$

Step 3.4.2

Divide each term in $- 7 x \leq - \frac{42}{5}$ by $- 7$ and simplify.

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Step 3.4.2.1

Divide each term in $- 7 x \leq - \frac{42}{5}$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \geq \frac{- \frac{42}{5}}{- 7}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} \geq \frac{- \frac{42}{5}}{- 7}$

Step 3.4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- \frac{42}{5}}{- 7}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \geq - \frac{42}{5} \cdot \frac{1}{- 7}$

Step 3.4.2.3.2

Cancel the common factor of $7$ .

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Step 3.4.2.3.2.1

Move the leading negative in $- \frac{42}{5}$ into the numerator.

$x \geq \frac{- 42}{5} \cdot \frac{1}{- 7}$

Step 3.4.2.3.2.2

Factor $7$ out of $- 42$ .

$x \geq \frac{7 (- 6)}{5} \cdot \frac{1}{- 7}$

Step 3.4.2.3.2.3

Factor $7$ out of $- 7$ .

$x \geq \frac{7 \cdot - 6}{5} \cdot \frac{1}{7 \cdot - 1}$

Step 3.4.2.3.2.4

Cancel the common factor.

$x \geq \frac{7 \cdot - 6}{5} \cdot \frac{1}{7 \cdot - 1}$

Step 3.4.2.3.2.5

Rewrite the expression.

$x \geq \frac{- 6}{5} \cdot \frac{1}{- 1}$

Step 3.4.2.3.3

Multiply $\frac{- 6}{5}$ by $\frac{1}{- 1}$ .

$x \geq \frac{- 6}{5 \cdot - 1}$

Step 3.4.2.3.4

Multiply $5$ by $- 1$ .

$x \geq \frac{- 6}{- 5}$

Step 3.4.2.3.5

Dividing two negative values results in a positive value.

$x \geq \frac{6}{5}$

Step 4

Find the union of the solutions.

$x \leq - \frac{12}{35}$ or $x \geq \frac{6}{5}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{12}{35} or x \geq \frac{6}{5}$

Interval Notation:

$(- \infty, - \frac{12}{35}] \cup [\frac{6}{5}, \infty)$

Step 6