Algebra Examples

$- \frac{1}{2} \leq \frac{4 - 3 x}{5} \leq \frac{1}{4}$

Step 1

Solve $- \frac{1}{2} \leq \frac{4 - 3 x}{5}$ for $x$ .

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Step 1.1

Multiply both sides by $5$ .

$- \frac{1}{2} \cdot 5 \leq \frac{4 - 3 x}{5} \cdot 5$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Simplify $- \frac{1}{2} \cdot 5$ .

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Step 1.2.1.1.1

Multiply $- \frac{1}{2} \cdot 5$ .

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Step 1.2.1.1.1.1

Multiply $5$ by $- 1$ .

$- 5 (\frac{1}{2}) \leq \frac{4 - 3 x}{5} \cdot 5$

Step 1.2.1.1.1.2

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{- 5}{2} \leq \frac{4 - 3 x}{5} \cdot 5$

Step 1.2.1.1.2

Move the negative in front of the fraction.

$- \frac{5}{2} \leq \frac{4 - 3 x}{5} \cdot 5$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Simplify $\frac{4 - 3 x}{5} \cdot 5$ .

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Step 1.2.2.1.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$- \frac{5}{2} \leq \frac{4 - 3 x}{5} \cdot 5$

Step 1.2.2.1.1.2

Rewrite the expression.

$- \frac{5}{2} \leq 4 - 3 x$

Step 1.2.2.1.2

Reorder $4$ and $- 3 x$ .

$- \frac{5}{2} \leq - 3 x + 4$

Step 1.3

Solve for $x$ .

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Step 1.3.1

Rewrite so $x$ is on the left side of the inequality.

$- 3 x + 4 \geq - \frac{5}{2}$

Step 1.3.2

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

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Step 1.3.2.1

Subtract $4$ from both sides of the inequality.

$- 3 x \geq - \frac{5}{2} - 4$

Step 1.3.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3 x \geq - \frac{5}{2} - 4 \cdot \frac{2}{2}$

Step 1.3.2.3

Combine $- 4$ and $\frac{2}{2}$ .

$- 3 x \geq - \frac{5}{2} + \frac{- 4 \cdot 2}{2}$

Step 1.3.2.4

Combine the numerators over the common denominator.

$- 3 x \geq \frac{- 5 - 4 \cdot 2}{2}$

Step 1.3.2.5

Simplify the numerator.

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Step 1.3.2.5.1

Multiply $- 4$ by $2$ .

$- 3 x \geq \frac{- 5 - 8}{2}$

Step 1.3.2.5.2

Subtract $8$ from $- 5$ .

$- 3 x \geq \frac{- 13}{2}$

Step 1.3.2.6

Move the negative in front of the fraction.

$- 3 x \geq - \frac{13}{2}$

Step 1.3.3

Divide each term in $- 3 x \geq - \frac{13}{2}$ by $- 3$ and simplify.

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Step 1.3.3.1

Divide each term in $- 3 x \geq - \frac{13}{2}$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{- \frac{13}{2}}{- 3}$

Step 1.3.3.2

Simplify the left side.

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Step 1.3.3.2.1

Cancel the common factor of $- 3$ .

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Step 1.3.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{- \frac{13}{2}}{- 3}$

Step 1.3.3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- \frac{13}{2}}{- 3}$

Step 1.3.3.3

Simplify the right side.

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Step 1.3.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq - \frac{13}{2} \cdot \frac{1}{- 3}$

Step 1.3.3.3.2

Move the negative in front of the fraction.

$x \leq - \frac{13}{2} (- \frac{1}{3})$

Step 1.3.3.3.3

Multiply $- \frac{13}{2} (- \frac{1}{3})$ .

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Step 1.3.3.3.3.1

Multiply $- 1$ by $- 1$ .

$x \leq 1 (\frac{13}{2}) \frac{1}{3}$

Step 1.3.3.3.3.2

Multiply $\frac{13}{2}$ by $1$ .

$x \leq \frac{13}{2} \cdot \frac{1}{3}$

Step 1.3.3.3.3.3

Multiply $\frac{13}{2}$ by $\frac{1}{3}$ .

$x \leq \frac{13}{2 \cdot 3}$

Step 1.3.3.3.3.4

Multiply $2$ by $3$ .

$x \leq \frac{13}{6}$

Step 2

Solve $\frac{4 - 3 x}{5} \leq \frac{1}{4}$ for $x$ .

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Step 2.1

Multiply both sides by $5$ .

$\frac{4 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

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 $\frac{4 - 3 x}{5} \cdot 5$ .

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Step 2.2.1.1.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.1.1

Cancel the common factor.

$\frac{4 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 2.2.1.1.1.2

Rewrite the expression.

$4 - 3 x \leq \frac{1}{4} \cdot 5$

Step 2.2.1.1.2

Reorder $4$ and $- 3 x$ .

$- 3 x + 4 \leq \frac{1}{4} \cdot 5$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Combine $\frac{1}{4}$ and $5$ .

$- 3 x + 4 \leq \frac{5}{4}$

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 $4$ from both sides of the inequality.

$- 3 x \leq \frac{5}{4} - 4$

Step 2.3.1.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 3 x \leq \frac{5}{4} - 4 \cdot \frac{4}{4}$

Step 2.3.1.3

Combine $- 4$ and $\frac{4}{4}$ .

$- 3 x \leq \frac{5}{4} + \frac{- 4 \cdot 4}{4}$

Step 2.3.1.4

Combine the numerators over the common denominator.

$- 3 x \leq \frac{5 - 4 \cdot 4}{4}$

Step 2.3.1.5

Simplify the numerator.

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Step 2.3.1.5.1

Multiply $- 4$ by $4$ .

$- 3 x \leq \frac{5 - 16}{4}$

Step 2.3.1.5.2

Subtract $16$ from $5$ .

$- 3 x \leq \frac{- 11}{4}$

Step 2.3.1.6

Move the negative in front of the fraction.

$- 3 x \leq - \frac{11}{4}$

Step 2.3.2

Divide each term in $- 3 x \leq - \frac{11}{4}$ by $- 3$ and simplify.

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Step 2.3.2.1

Divide each term in $- 3 x \leq - \frac{11}{4}$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- \frac{11}{4}}{- 3}$

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.

$x \geq - \frac{11}{4} \cdot \frac{1}{- 3}$

Step 2.3.2.3.2

Move the negative in front of the fraction.

$x \geq - \frac{11}{4} (- \frac{1}{3})$

Step 2.3.2.3.3

Multiply $- \frac{11}{4} (- \frac{1}{3})$ .

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Step 2.3.2.3.3.1

Multiply $- 1$ by $- 1$ .

$x \geq 1 (\frac{11}{4}) \frac{1}{3}$

Step 2.3.2.3.3.2

Multiply $\frac{11}{4}$ by $1$ .

$x \geq \frac{11}{4} \cdot \frac{1}{3}$

Step 2.3.2.3.3.3

Multiply $\frac{11}{4}$ by $\frac{1}{3}$ .

$x \geq \frac{11}{4 \cdot 3}$

Step 2.3.2.3.3.4

Multiply $4$ by $3$ .

$x \geq \frac{11}{12}$

Step 3

Find the intersection of $x \leq \frac{13}{6}$ and $x \geq \frac{11}{12}$ .

$\frac{11}{12} \leq x \leq \frac{13}{6}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$\frac{11}{12} \leq x \leq \frac{13}{6}$

Interval Notation:

$[\frac{11}{12}, \frac{13}{6}]$

Step 5