Precalculus Examples

$\frac{2 x}{5} - \frac{1}{2} \cdot (x - 3) < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{2 x}{5} - \frac{1}{2} x - \frac{1}{2} \cdot - 3 < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.1.2

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

$\frac{2 x}{5} - \frac{x}{2} - \frac{1}{2} \cdot - 3 < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.1.3

Multiply $- \frac{1}{2} \cdot - 3$ .

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Step 1.1.3.1

Multiply $- 3$ by $- 1$ .

$\frac{2 x}{5} - \frac{x}{2} + 3 (\frac{1}{2}) < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.1.3.2

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

$\frac{2 x}{5} - \frac{x}{2} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.2

To write $\frac{2 x}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 x}{5} \cdot \frac{2}{2} - \frac{x}{2} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.3

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 x}{5} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{5}{5} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 x}{5}$ by $\frac{2}{2}$ .

$\frac{2 x \cdot 2}{5 \cdot 2} - \frac{x}{2} \cdot \frac{5}{5} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.4.2

Multiply $5$ by $2$ .

$\frac{2 x \cdot 2}{10} - \frac{x}{2} \cdot \frac{5}{5} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.4.3

Multiply $\frac{x}{2}$ by $\frac{5}{5}$ .

$\frac{2 x \cdot 2}{10} - \frac{x \cdot 5}{2 \cdot 5} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.4.4

Multiply $2$ by $5$ .

$\frac{2 x \cdot 2}{10} - \frac{x \cdot 5}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - x \cdot 5}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $x$ out of $2 x \cdot 2 - x \cdot 5$ .

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Step 1.6.1.1.1

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

$\frac{x (2 \cdot 2) - x \cdot 5}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.1.1.2

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

$\frac{x (2 \cdot 2) + x (- 1 \cdot 5)}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.1.1.3

Factor $x$ out of $x (2 \cdot 2) + x (- 1 \cdot 5)$ .

$\frac{x (2 \cdot 2 - 1 \cdot 5)}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.1.2

Multiply $2$ by $2$ .

$\frac{x (4 - 1 \cdot 5)}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.1.3

Multiply $- 1$ by $5$ .

$\frac{x (4 - 5)}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.1.4

Subtract $5$ from $4$ .

$\frac{x \cdot - 1}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.2

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

$\frac{- 1 \cdot x}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 1.6.3

Move the negative in front of the fraction.

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - (x + 2)$

Step 2

Simplify $\frac{2 x}{3} - \frac{3}{10} - (x + 2)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - x - 1 \cdot 2$

Step 2.1.2

Multiply $- 1$ by $2$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x}{3} - \frac{3}{10} - x - 2$

Step 2.2

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

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x}{3} - x \cdot \frac{3}{3} - \frac{3}{10} - 2$

Step 2.3

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

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x}{3} + \frac{- x \cdot 3}{3} - \frac{3}{10} - 2$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x - x \cdot 3}{3} - \frac{3}{10} - 2$

Step 2.5

Find the common denominator.

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Step 2.5.1

Multiply $\frac{2 x - x \cdot 3}{3}$ by $\frac{10}{10}$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{2 x - x \cdot 3}{3} \cdot \frac{10}{10} - \frac{3}{10} - 2$

Step 2.5.2

Multiply $\frac{2 x - x \cdot 3}{3}$ by $\frac{10}{10}$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{3 \cdot 10} - \frac{3}{10} - 2$

Step 2.5.3

Multiply $\frac{3}{10}$ by $\frac{3}{3}$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{3 \cdot 10} - (\frac{3}{10} \cdot \frac{3}{3}) - 2$

Step 2.5.4

Multiply $\frac{3}{10}$ by $\frac{3}{3}$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{3 \cdot 10} - \frac{3 \cdot 3}{10 \cdot 3} - 2$

Step 2.5.5

Write $- 2$ as a fraction with denominator $1$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{3 \cdot 10} - \frac{3 \cdot 3}{10 \cdot 3} + \frac{- 2}{1}$

Step 2.5.6

Multiply $\frac{- 2}{1}$ by $\frac{30}{30}$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{3 \cdot 10} - \frac{3 \cdot 3}{10 \cdot 3} + \frac{- 2}{1} \cdot \frac{30}{30}$

Step 2.5.7

Multiply $\frac{- 2}{1}$ by $\frac{30}{30}$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{3 \cdot 10} - \frac{3 \cdot 3}{10 \cdot 3} + \frac{- 2 \cdot 30}{30}$

Step 2.5.8

Multiply $3$ by $10$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{30} - \frac{3 \cdot 3}{10 \cdot 3} + \frac{- 2 \cdot 30}{30}$

Step 2.5.9

Reorder the factors of $10 \cdot 3$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{30} - \frac{3 \cdot 3}{3 \cdot 10} + \frac{- 2 \cdot 30}{30}$

Step 2.5.10

Multiply $3$ by $10$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10}{30} - \frac{3 \cdot 3}{30} + \frac{- 2 \cdot 30}{30}$

Step 2.6

Combine the numerators over the common denominator.

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - x \cdot 3) \cdot 10 - 3 \cdot 3 - 2 \cdot 30}{30}$

Step 2.7

Simplify each term.

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Step 2.7.1

Multiply $3$ by $- 1$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{(2 x - 3 x) \cdot 10 - 3 \cdot 3 - 2 \cdot 30}{30}$

Step 2.7.2

Subtract $3 x$ from $2 x$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- x \cdot 10 - 3 \cdot 3 - 2 \cdot 30}{30}$

Step 2.7.3

Multiply $10$ by $- 1$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- 10 x - 3 \cdot 3 - 2 \cdot 30}{30}$

Step 2.7.4

Multiply $- 3$ by $3$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- 10 x - 9 - 2 \cdot 30}{30}$

Step 2.7.5

Multiply $- 2$ by $30$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- 10 x - 9 - 60}{30}$

Step 2.8

Simplify with factoring out.

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Step 2.8.1

Subtract $60$ from $- 9$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- 10 x - 69}{30}$

Step 2.8.2

Factor $- 1$ out of $- 10 x$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- (10 x) - 69}{30}$

Step 2.8.3

Rewrite $- 69$ as $- 1 (69)$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- (10 x) - 1 (69)}{30}$

Step 2.8.4

Factor $- 1$ out of $- (10 x) - 1 (69)$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- (10 x + 69)}{30}$

Step 2.8.5

Simplify the expression.

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Step 2.8.5.1

Rewrite $- (10 x + 69)$ as $- 1 (10 x + 69)$ .

$- \frac{x}{10} + \frac{3}{2} < \frac{- 1 (10 x + 69)}{30}$

Step 2.8.5.2

Move the negative in front of the fraction.

$- \frac{x}{10} + \frac{3}{2} < - \frac{10 x + 69}{30}$

Step 3

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

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Step 3.1

Add $\frac{10 x + 69}{30}$ to both sides of the inequality.

$- \frac{x}{10} + \frac{3}{2} + \frac{10 x + 69}{30} < 0$

Step 3.2

To write $- \frac{x}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{x}{10} \cdot \frac{3}{3} + \frac{10 x + 69}{30} + \frac{3}{2} < 0$

Step 3.3

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{x}{10}$ by $\frac{3}{3}$ .

$- \frac{x \cdot 3}{10 \cdot 3} + \frac{10 x + 69}{30} + \frac{3}{2} < 0$

Step 3.3.2

Multiply $10$ by $3$ .

$- \frac{x \cdot 3}{30} + \frac{10 x + 69}{30} + \frac{3}{2} < 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- x \cdot 3 + 10 x + 69}{30} + \frac{3}{2} < 0$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3$ by $- 1$ .

$\frac{- 3 x + 10 x + 69}{30} + \frac{3}{2} < 0$

Step 3.5.2

Add $- 3 x$ and $10 x$ .

$\frac{7 x + 69}{30} + \frac{3}{2} < 0$

Step 3.6

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{7 x + 69}{30} + \frac{3}{2} \cdot \frac{15}{15} < 0$

Step 3.7

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.1

Multiply $\frac{3}{2}$ by $\frac{15}{15}$ .

$\frac{7 x + 69}{30} + \frac{3 \cdot 15}{2 \cdot 15} < 0$

Step 3.7.2

Multiply $2$ by $15$ .

$\frac{7 x + 69}{30} + \frac{3 \cdot 15}{30} < 0$

Step 3.8

Combine the numerators over the common denominator.

$\frac{7 x + 69 + 3 \cdot 15}{30} < 0$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $3$ by $15$ .

$\frac{7 x + 69 + 45}{30} < 0$

Step 3.9.2

Add $69$ and $45$ .

$\frac{7 x + 114}{30} < 0$

Step 4

Multiply both sides by $30$ .

$\frac{7 x + 114}{30} \cdot 30 < 0 \cdot 30$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $30$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{7 x + 114}{30} \cdot 30 < 0 \cdot 30$

Step 5.1.1.2

Rewrite the expression.

$7 x + 114 < 0 \cdot 30$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $0$ by $30$ .

$7 x + 114 < 0$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $114$ from both sides of the inequality.

$7 x < - 114$

Step 6.2

Divide each term in $7 x < - 114$ by $7$ and simplify.

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Step 6.2.1

Divide each term in $7 x < - 114$ by $7$ .

$\frac{7 x}{7} < \frac{- 114}{7}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $7$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 114}{7}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 114}{7}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Move the negative in front of the fraction.

$x < - \frac{114}{7}$

Step 7

Convert the inequality to interval notation.

$(- \infty, - \frac{114}{7})$

Step 8