Algebra Examples

$7 - \frac{2}{3} x < x - 8$

Step 1

Simplify each term.

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Step 1.1

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

$7 - \frac{x \cdot 2}{3} < x - 8$

Step 1.2

Move $2$ to the left of $x$ .

$7 - \frac{2 x}{3} < x - 8$

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.

$7 - \frac{2 x}{3} - x < - 8$

Step 2.2

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

$7 - \frac{2 x}{3} - x \cdot \frac{3}{3} < - 8$

Step 2.3

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

$7 - \frac{2 x}{3} + \frac{- x \cdot 3}{3} < - 8$

Step 2.4

Combine the numerators over the common denominator.

$7 + \frac{- 2 x - x \cdot 3}{3} < - 8$

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 $- 2 x - x \cdot 3$ .

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Step 2.5.1.1.1

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

$7 + \frac{x \cdot - 2 - x \cdot 3}{3} < - 8$

Step 2.5.1.1.2

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

$7 + \frac{x \cdot - 2 + x (- 1 \cdot 3)}{3} < - 8$

Step 2.5.1.1.3

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

$7 + \frac{x (- 2 - 1 \cdot 3)}{3} < - 8$

Step 2.5.1.2

Multiply $- 1$ by $3$ .

$7 + \frac{x (- 2 - 3)}{3} < - 8$

Step 2.5.1.3

Subtract $3$ from $- 2$ .

$7 + \frac{x \cdot - 5}{3} < - 8$

Step 2.5.2

Move $- 5$ to the left of $x$ .

$7 + \frac{- 5 \cdot x}{3} < - 8$

Step 2.5.3

Move the negative in front of the fraction.

$7 - \frac{5 x}{3} < - 8$

Step 3

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

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Step 3.1

Subtract $7$ from both sides of the inequality.

$- \frac{5 x}{3} < - 8 - 7$

Step 3.2

Subtract $7$ from $- 8$ .

$- \frac{5 x}{3} < - 15$

Step 4

Divide each term in $- \frac{5 x}{3} < - 15$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{5 x}{3} < - 15$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 x}{3}}{- 1} > \frac{- 15}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 x}{3}}{1} > \frac{- 15}{- 1}$

Step 4.2.2

Divide $\frac{5 x}{3}$ by $1$ .

$\frac{5 x}{3} > \frac{- 15}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 15$ by $- 1$ .

$\frac{5 x}{3} > 15$

Step 5

Multiply both sides by $3$ .

$\frac{5 x}{3} \cdot 3 > 15 \cdot 3$

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 $3$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{5 x}{3} \cdot 3 > 15 \cdot 3$

Step 6.1.1.2

Rewrite the expression.

$5 x > 15 \cdot 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $15$ by $3$ .

$5 x > 45$

Step 7

Divide each term in $5 x > 45$ by $5$ and simplify.

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Step 7.1

Divide each term in $5 x > 45$ by $5$ .

$\frac{5 x}{5} > \frac{45}{5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{45}{5}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x > \frac{45}{5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $45$ by $5$ .

$x > 9$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > 9$

Interval Notation:

$(9, \infty)$