Algebra Examples

$\frac{5}{6} (3 - x) - \frac{1}{2} (x - 4) \geq \frac{1}{3} (2 x - 3) - x$

Step 1

Simplify $\frac{5}{6} \cdot (3 - x) - \frac{1}{2} \cdot (x - 4)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{5}{6} \cdot 3 + \frac{5}{6} (- x) - \frac{1}{2} \cdot (x - 4) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.2

Cancel the common factor of $3$ .

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Step 1.1.2.1

Factor $3$ out of $6$ .

$\frac{5}{3 (2)} \cdot 3 + \frac{5}{6} (- x) - \frac{1}{2} \cdot (x - 4) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.2.2

Cancel the common factor.

$\frac{5}{3 \cdot 2} \cdot 3 + \frac{5}{6} (- x) - \frac{1}{2} \cdot (x - 4) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.2.3

Rewrite the expression.

$\frac{5}{2} + \frac{5}{6} (- x) - \frac{1}{2} \cdot (x - 4) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.3

Combine $\frac{5}{6}$ and $x$ .

$\frac{5}{2} - \frac{5 x}{6} - \frac{1}{2} \cdot (x - 4) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.4

Apply the distributive property.

$\frac{5}{2} - \frac{5 x}{6} - \frac{1}{2} x - \frac{1}{2} \cdot - 4 \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.5

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

$\frac{5}{2} - \frac{5 x}{6} - \frac{x}{2} - \frac{1}{2} \cdot - 4 \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.6

Cancel the common factor of $2$ .

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Step 1.1.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{5}{2} - \frac{5 x}{6} - \frac{x}{2} + \frac{- 1}{2} \cdot - 4 \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.6.2

Factor $2$ out of $- 4$ .

$\frac{5}{2} - \frac{5 x}{6} - \frac{x}{2} + \frac{- 1}{2} \cdot (2 (- 2)) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.6.3

Cancel the common factor.

$\frac{5}{2} - \frac{5 x}{6} - \frac{x}{2} + \frac{- 1}{2} \cdot (2 \cdot - 2) \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.6.4

Rewrite the expression.

$\frac{5}{2} - \frac{5 x}{6} - \frac{x}{2} - 1 \cdot - 2 \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.1.7

Multiply $- 1$ by $- 2$ .

$\frac{5}{2} - \frac{5 x}{6} - \frac{x}{2} + 2 \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.2

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

$- \frac{5 x}{6} - \frac{x}{2} + \frac{5}{2} + 2 \cdot \frac{2}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.3

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

$- \frac{5 x}{6} - \frac{x}{2} + \frac{5}{2} + \frac{2 \cdot 2}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.4

Combine the numerators over the common denominator.

$- \frac{5 x}{6} - \frac{x}{2} + \frac{5 + 2 \cdot 2}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $2$ by $2$ .

$- \frac{5 x}{6} - \frac{x}{2} + \frac{5 + 4}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.5.2

Add $5$ and $4$ .

$- \frac{5 x}{6} - \frac{x}{2} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.6

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

$- \frac{5 x}{6} - \frac{x}{2} \cdot \frac{3}{3} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.7

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

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Step 1.7.1

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

$- \frac{5 x}{6} - \frac{x \cdot 3}{2 \cdot 3} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.7.2

Multiply $2$ by $3$ .

$- \frac{5 x}{6} - \frac{x \cdot 3}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.8

Combine the numerators over the common denominator.

$\frac{- 5 x - x \cdot 3}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9

Simplify each term.

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Step 1.9.1

Simplify the numerator.

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Step 1.9.1.1

Factor $x$ out of $- 5 x - x \cdot 3$ .

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Step 1.9.1.1.1

Factor $x$ out of $- 5 x$ .

$\frac{x \cdot - 5 - x \cdot 3}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.1.1.2

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

$\frac{x \cdot - 5 + x (- 1 \cdot 3)}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.1.1.3

Factor $x$ out of $x \cdot - 5 + x (- 1 \cdot 3)$ .

$\frac{x (- 5 - 1 \cdot 3)}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.1.2

Multiply $- 1$ by $3$ .

$\frac{x (- 5 - 3)}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.1.3

Subtract $3$ from $- 5$ .

$\frac{x \cdot - 8}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.2

Cancel the common factor of $- 8$ and $6$ .

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Step 1.9.2.1

Factor $2$ out of $x \cdot - 8$ .

$\frac{2 (x \cdot - 4)}{6} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.2.2

Cancel the common factors.

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Step 1.9.2.2.1

Factor $2$ out of $6$ .

$\frac{2 (x \cdot - 4)}{2 (3)} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.2.2.2

Cancel the common factor.

$\frac{2 (x \cdot - 4)}{2 \cdot 3} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.2.2.3

Rewrite the expression.

$\frac{x \cdot - 4}{3} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.3

Move $- 4$ to the left of $x$ .

$\frac{- 4 \cdot x}{3} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 1.9.4

Move the negative in front of the fraction.

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{1}{3} \cdot (2 x - 3) - x$

Step 2

Simplify $\frac{1}{3} \cdot (2 x - 3) - x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{1}{3} (2 x) + \frac{1}{3} \cdot - 3 - x$

Step 2.1.2

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

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Step 2.1.2.1

Combine $2$ and $\frac{1}{3}$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2}{3} x + \frac{1}{3} \cdot - 3 - x$

Step 2.1.2.2

Combine $\frac{2}{3}$ and $x$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x}{3} + \frac{1}{3} \cdot - 3 - x$

Step 2.1.3

Cancel the common factor of $3$ .

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Step 2.1.3.1

Factor $3$ out of $- 3$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x}{3} + \frac{1}{3} \cdot (3 (- 1)) - x$

Step 2.1.3.2

Cancel the common factor.

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x}{3} + \frac{1}{3} \cdot (3 \cdot - 1) - x$

Step 2.1.3.3

Rewrite the expression.

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x}{3} - 1 - x$

Step 2.2

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

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x}{3} - x \cdot \frac{3}{3} - 1$

Step 2.3

Simplify terms.

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Step 2.3.1

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

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x}{3} + \frac{- x \cdot 3}{3} - 1$

Step 2.3.2

Combine the numerators over the common denominator.

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{2 x - x \cdot 3}{3} - 1$

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor $x$ out of $2 x - x \cdot 3$ .

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Step 2.4.1.1.1

Factor $x$ out of $2 x$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{x \cdot 2 - x \cdot 3}{3} - 1$

Step 2.4.1.1.2

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

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{x \cdot 2 + x (- 1 \cdot 3)}{3} - 1$

Step 2.4.1.1.3

Factor $x$ out of $x \cdot 2 + x (- 1 \cdot 3)$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{x (2 - 1 \cdot 3)}{3} - 1$

Step 2.4.1.2

Multiply $- 1$ by $3$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{x (2 - 3)}{3} - 1$

Step 2.4.1.3

Subtract $3$ from $2$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{x \cdot - 1}{3} - 1$

Step 2.4.2

Move $- 1$ to the left of $x$ .

$- \frac{4 x}{3} + \frac{9}{2} \geq \frac{- 1 \cdot x}{3} - 1$

Step 2.4.3

Move the negative in front of the fraction.

$- \frac{4 x}{3} + \frac{9}{2} \geq - \frac{x}{3} - 1$

Step 3

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

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Step 3.1

Add $\frac{x}{3}$ to both sides of the inequality.

$- \frac{4 x}{3} + \frac{9}{2} + \frac{x}{3} \geq - 1$

Step 3.2

Combine the numerators over the common denominator.

$\frac{- 4 x + x}{3} + \frac{9}{2} \geq - 1$

Step 3.3

Add $- 4 x$ and $x$ .

$\frac{- 3 x}{3} + \frac{9}{2} \geq - 1$

Step 3.4

Cancel the common factor of $- 3$ and $3$ .

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Step 3.4.1

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

$\frac{3 (- x)}{3} + \frac{9}{2} \geq - 1$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (- x)}{3 (1)} + \frac{9}{2} \geq - 1$

Step 3.4.2.2

Cancel the common factor.

$\frac{3 (- x)}{3 \cdot 1} + \frac{9}{2} \geq - 1$

Step 3.4.2.3

Rewrite the expression.

$\frac{- x}{1} + \frac{9}{2} \geq - 1$

Step 3.4.2.4

Divide $- x$ by $1$ .

$- x + \frac{9}{2} \geq - 1$

Step 4

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

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Step 4.1

Subtract $\frac{9}{2}$ from both sides of the inequality.

$- x \geq - 1 - \frac{9}{2}$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x \geq - 1 \cdot \frac{2}{2} - \frac{9}{2}$

Step 4.3

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

$- x \geq \frac{- 1 \cdot 2}{2} - \frac{9}{2}$

Step 4.4

Combine the numerators over the common denominator.

$- x \geq \frac{- 1 \cdot 2 - 9}{2}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$- x \geq \frac{- 2 - 9}{2}$

Step 4.5.2

Subtract $9$ from $- 2$ .

$- x \geq \frac{- 11}{2}$

Step 4.6

Move the negative in front of the fraction.

$- x \geq - \frac{11}{2}$

Step 5

Divide each term in $- x \geq - \frac{11}{2}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x \geq - \frac{11}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- \frac{11}{2}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- \frac{11}{2}}{- 1}$

Step 5.2.2

Divide $x$ by $1$ .

$x \leq \frac{- \frac{11}{2}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{\frac{11}{2}}{1}$

Step 5.3.2

Divide $\frac{11}{2}$ by $1$ .

$x \leq \frac{11}{2}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{11}{2}$

Interval Notation:

$(- \infty, \frac{11}{2}]$