Algebra Examples

$\frac{7}{3} - \frac{1}{2} x \geq \frac{7}{6} + x$

Step 1

Combine $x$ and $\frac{1}{2}$ .

$\frac{7}{3} - \frac{x}{2} \geq \frac{7}{6} + x$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the inequality.

$\frac{7}{3} - \frac{x}{2} - x \geq \frac{7}{6}$

Step 2.2

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

$\frac{7}{3} - \frac{x}{2} - x \cdot \frac{2}{2} \geq \frac{7}{6}$

Step 2.3

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

$\frac{7}{3} - \frac{x}{2} + \frac{- x \cdot 2}{2} \geq \frac{7}{6}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{7}{3} + \frac{- x - x \cdot 2}{2} \geq \frac{7}{6}$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $x$ out of $- x - x \cdot 2$ .

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Step 2.5.1.1.1

Factor $x$ out of $- x$ .

$\frac{7}{3} + \frac{x \cdot - 1 - x \cdot 2}{2} \geq \frac{7}{6}$

Step 2.5.1.1.2

Factor $x$ out of $- x \cdot 2$ .

$\frac{7}{3} + \frac{x \cdot - 1 + x (- 1 \cdot 2)}{2} \geq \frac{7}{6}$

Step 2.5.1.1.3

Factor $x$ out of $x \cdot - 1 + x (- 1 \cdot 2)$ .

$\frac{7}{3} + \frac{x (- 1 - 1 \cdot 2)}{2} \geq \frac{7}{6}$

Step 2.5.1.2

Multiply $- 1$ by $2$ .

$\frac{7}{3} + \frac{x (- 1 - 2)}{2} \geq \frac{7}{6}$

Step 2.5.1.3

Subtract $2$ from $- 1$ .

$\frac{7}{3} + \frac{x \cdot - 3}{2} \geq \frac{7}{6}$

Step 2.5.2

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

$\frac{7}{3} + \frac{- 3 \cdot x}{2} \geq \frac{7}{6}$

Step 2.5.3

Move the negative in front of the fraction.

$\frac{7}{3} - \frac{3 x}{2} \geq \frac{7}{6}$

Step 3

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

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Step 3.1

Subtract $\frac{7}{3}$ from both sides of the inequality.

$- \frac{3 x}{2} \geq \frac{7}{6} - \frac{7}{3}$

Step 3.2

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 x}{2} \geq \frac{7}{6} - \frac{7}{3} \cdot \frac{2}{2}$

Step 3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{7}{3}$ by $\frac{2}{2}$ .

$- \frac{3 x}{2} \geq \frac{7}{6} - \frac{7 \cdot 2}{3 \cdot 2}$

Step 3.3.2

Multiply $3$ by $2$ .

$- \frac{3 x}{2} \geq \frac{7}{6} - \frac{7 \cdot 2}{6}$

Step 3.4

Combine the numerators over the common denominator.

$- \frac{3 x}{2} \geq \frac{7 - 7 \cdot 2}{6}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 7$ by $2$ .

$- \frac{3 x}{2} \geq \frac{7 - 14}{6}$

Step 3.5.2

Subtract $14$ from $7$ .

$- \frac{3 x}{2} \geq \frac{- 7}{6}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{3 x}{2} \geq - \frac{7}{6}$

Step 4

Divide each term in $- \frac{3 x}{2} \geq - \frac{7}{6}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{3 x}{2} \geq - \frac{7}{6}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 x}{2}}{- 1} \leq \frac{- \frac{7}{6}}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{3 x}{2}}{1} \leq \frac{- \frac{7}{6}}{- 1}$

Step 4.2.2

Divide $\frac{3 x}{2}$ by $1$ .

$\frac{3 x}{2} \leq \frac{- \frac{7}{6}}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Dividing two negative values results in a positive value.

$\frac{3 x}{2} \leq \frac{\frac{7}{6}}{1}$

Step 4.3.2

Divide $\frac{7}{6}$ by $1$ .

$\frac{3 x}{2} \leq \frac{7}{6}$

Step 5

Multiply both sides by $2$ .

$\frac{3 x}{2} \cdot 2 \leq \frac{7}{6} \cdot 2$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{3 x}{2} \cdot 2 \leq \frac{7}{6} \cdot 2$

Step 6.1.1.2

Rewrite the expression.

$3 x \leq \frac{7}{6} \cdot 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Factor $2$ out of $6$ .

$3 x \leq \frac{7}{2 (3)} \cdot 2$

Step 6.2.1.2

Cancel the common factor.

$3 x \leq \frac{7}{2 \cdot 3} \cdot 2$

Step 6.2.1.3

Rewrite the expression.

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

Step 7

Divide each term in $3 x \leq \frac{7}{3}$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 x \leq \frac{7}{3}$ by $3$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{7}{3} \cdot \frac{1}{3}$

Step 7.3.2

Multiply $\frac{7}{3} \cdot \frac{1}{3}$ .

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Step 7.3.2.1

Multiply $\frac{7}{3}$ by $\frac{1}{3}$ .

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

Step 7.3.2.2

Multiply $3$ by $3$ .

$x \leq \frac{7}{9}$

Step 8