Pre-Algebra Examples

$- \frac{5}{9} x - \frac{1}{9} < \frac{1}{3}$

Step 1

Simplify each term.

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Step 1.1

Combine $x$ and $\frac{5}{9}$ .

$- \frac{x \cdot 5}{9} - \frac{1}{9} < \frac{1}{3}$

Step 1.2

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

$- \frac{5 x}{9} - \frac{1}{9} < \frac{1}{3}$

Step 2

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

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Step 2.1

Add $\frac{1}{9}$ to both sides of the inequality.

$- \frac{5 x}{9} < \frac{1}{3} + \frac{1}{9}$

Step 2.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{5 x}{9} < \frac{1}{3} \cdot \frac{3}{3} + \frac{1}{9}$

Step 2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

$- \frac{5 x}{9} < \frac{3}{3 \cdot 3} + \frac{1}{9}$

Step 2.3.2

Multiply $3$ by $3$ .

$- \frac{5 x}{9} < \frac{3}{9} + \frac{1}{9}$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{5 x}{9} < \frac{3 + 1}{9}$

Step 2.5

Add $3$ and $1$ .

$- \frac{5 x}{9} < \frac{4}{9}$

Step 3

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

$- 5 x < 4$

Step 4

Divide each term in $- 5 x < 4$ by $- 5$ and simplify.

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Step 4.1

Divide each term in $- 5 x < 4$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{4}{- 5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{4}{- 5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x > \frac{4}{- 5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x > - \frac{4}{5}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > - \frac{4}{5}$

Interval Notation:

$(- \frac{4}{5}, \infty)$