Algebra Examples

$- \frac{3}{4} x \geq - \frac{11}{12} + \frac{1}{3} x$

Step 1

Solve $x \leq \frac{11}{13}$ .

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Step 1.1

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

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Step 1.1.1

Rewrite.

$0 + 0 - \frac{3}{4} x \geq - \frac{11}{12} + \frac{1}{3} x$

Step 1.1.2

Simplify by adding zeros.

$- \frac{3}{4} x \geq - \frac{11}{12} + \frac{1}{3} x$

Step 1.1.3

Combine $x$ and $\frac{3}{4}$ .

$- \frac{x \cdot 3}{4} \geq - \frac{11}{12} + \frac{1}{3} x$

Step 1.1.4

Move $3$ to the left of $x$ .

$- \frac{3 x}{4} \geq - \frac{11}{12} + \frac{1}{3} x$

Step 1.2

Combine $\frac{1}{3}$ and $x$ .

$- \frac{3 x}{4} \geq - \frac{11}{12} + \frac{x}{3}$

Step 1.3

Move all terms containing $x$ to the left side of the inequality.

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Step 1.3.1

Subtract $\frac{x}{3}$ from both sides of the inequality.

$- \frac{3 x}{4} - \frac{x}{3} \geq - \frac{11}{12}$

Step 1.3.2

To write $- \frac{3 x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{3 x}{4} \cdot \frac{3}{3} - \frac{x}{3} \geq - \frac{11}{12}$

Step 1.3.3

To write $- \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3 x}{4} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{4}{4} \geq - \frac{11}{12}$

Step 1.3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.4.1

Multiply $\frac{3 x}{4}$ by $\frac{3}{3}$ .

$- \frac{3 x \cdot 3}{4 \cdot 3} - \frac{x}{3} \cdot \frac{4}{4} \geq - \frac{11}{12}$

Step 1.3.4.2

Multiply $4$ by $3$ .

$- \frac{3 x \cdot 3}{12} - \frac{x}{3} \cdot \frac{4}{4} \geq - \frac{11}{12}$

Step 1.3.4.3

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$- \frac{3 x \cdot 3}{12} - \frac{x \cdot 4}{3 \cdot 4} \geq - \frac{11}{12}$

Step 1.3.4.4

Multiply $3$ by $4$ .

$- \frac{3 x \cdot 3}{12} - \frac{x \cdot 4}{12} \geq - \frac{11}{12}$

Step 1.3.5

Combine the numerators over the common denominator.

$\frac{- 3 x \cdot 3 - x \cdot 4}{12} \geq - \frac{11}{12}$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Factor $x$ out of $- 3 x \cdot 3 - x \cdot 4$ .

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Step 1.3.6.1.1

Factor $x$ out of $- 3 x \cdot 3$ .

$\frac{x (- 3 \cdot 3) - x \cdot 4}{12} \geq - \frac{11}{12}$

Step 1.3.6.1.2

Factor $x$ out of $- x \cdot 4$ .

$\frac{x (- 3 \cdot 3) + x (- 1 \cdot 4)}{12} \geq - \frac{11}{12}$

Step 1.3.6.1.3

Factor $x$ out of $x (- 3 \cdot 3) + x (- 1 \cdot 4)$ .

$\frac{x (- 3 \cdot 3 - 1 \cdot 4)}{12} \geq - \frac{11}{12}$

Step 1.3.6.2

Multiply $- 3$ by $3$ .

$\frac{x (- 9 - 1 \cdot 4)}{12} \geq - \frac{11}{12}$

Step 1.3.6.3

Multiply $- 1$ by $4$ .

$\frac{x (- 9 - 4)}{12} \geq - \frac{11}{12}$

Step 1.3.6.4

Subtract $4$ from $- 9$ .

$\frac{x \cdot - 13}{12} \geq - \frac{11}{12}$

Step 1.3.7

Move $- 13$ to the left of $x$ .

$\frac{- 13 \cdot x}{12} \geq - \frac{11}{12}$

Step 1.3.8

Move the negative in front of the fraction.

$- \frac{13 x}{12} \geq - \frac{11}{12}$

Step 1.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 13 x \geq - 11$

Step 1.5

Divide each term in $- 13 x \geq - 11$ by $- 13$ and simplify.

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Step 1.5.1

Divide each term in $- 13 x \geq - 11$ by $- 13$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 13 x}{- 13} \leq \frac{- 11}{- 13}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $- 13$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{- 13 x}{- 13} \leq \frac{- 11}{- 13}$

Step 1.5.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 11}{- 13}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Dividing two negative values results in a positive value.

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

Step 2

Use the inequality $x \leq \frac{11}{13}$ to build the set notation.

${x | x \leq \frac{11}{13}}$

Step 3