Precalculus Examples

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

Step 1

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

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Step 1.1

Rewrite.

$3 x + 4 > 0 + 0 + \frac{1}{3} \cdot (x - 2)$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

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

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

Step 1.5

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

$3 x + 4 > \frac{x}{3} + \frac{- 2}{3}$

Step 1.6

Move the negative in front of the fraction.

$3 x + 4 > \frac{x}{3} - \frac{2}{3}$

Step 2

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

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Step 2.1

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

$3 x + 4 - \frac{x}{3} > - \frac{2}{3}$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

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 $3 x \cdot 3 - x$ .

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Step 2.5.1.1.1

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

$\frac{x (3 \cdot 3) - x}{3} + 4 > - \frac{2}{3}$

Step 2.5.1.1.2

Factor $x$ out of $- x$ .

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

Step 2.5.1.1.3

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

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

Step 2.5.1.2

Multiply $3$ by $3$ .

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

Step 2.5.1.3

Subtract $1$ from $9$ .

$\frac{x \cdot 8}{3} + 4 > - \frac{2}{3}$

Step 2.5.2

Move $8$ to the left of $x$ .

$\frac{8 x}{3} + 4 > - \frac{2}{3}$

Step 3

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

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Step 3.1

Subtract $4$ from both sides of the inequality.

$\frac{8 x}{3} > - \frac{2}{3} - 4$

Step 3.2

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

$\frac{8 x}{3} > - \frac{2}{3} - 4 \cdot \frac{3}{3}$

Step 3.3

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

$\frac{8 x}{3} > - \frac{2}{3} + \frac{- 4 \cdot 3}{3}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{8 x}{3} > \frac{- 2 - 4 \cdot 3}{3}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 4$ by $3$ .

$\frac{8 x}{3} > \frac{- 2 - 12}{3}$

Step 3.5.2

Subtract $12$ from $- 2$ .

$\frac{8 x}{3} > \frac{- 14}{3}$

Step 3.6

Move the negative in front of the fraction.

$\frac{8 x}{3} > - \frac{14}{3}$

Step 4

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

$8 x > - 14$

Step 5

Divide each term in $8 x > - 14$ by $8$ and simplify.

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Step 5.1

Divide each term in $8 x > - 14$ by $8$ .

$\frac{8 x}{8} > \frac{- 14}{8}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $8$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{8 x}{8} > \frac{- 14}{8}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 14}{8}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 14$ and $8$ .

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Step 5.3.1.1

Factor $2$ out of $- 14$ .

$x > \frac{2 (- 7)}{8}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $8$ .

$x > \frac{2 \cdot - 7}{2 \cdot 4}$

Step 5.3.1.2.2

Cancel the common factor.

$x > \frac{2 \cdot - 7}{2 \cdot 4}$

Step 5.3.1.2.3

Rewrite the expression.

$x > \frac{- 7}{4}$

Step 5.3.2

Move the negative in front of the fraction.

$x > - \frac{7}{4}$

Step 6

Convert the inequality to interval notation.

$(- \frac{7}{4}, \infty)$

Step 7