Algebra Examples

$\frac{9 - 8 w}{5} < 3$ or $\frac{2}{3} (18 w - 3) < - 62$

Step 1

Simplify the first inequality.

Tap for more steps...

Step 1.1

Multiply both sides by $5$ .

$\frac{9 - 8 w}{5} \cdot 5 < 3 \cdot 5$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.1.1

Simplify $\frac{9 - 8 w}{5} \cdot 5$ .

Tap for more steps...

Step 1.2.1.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.2.1.1.1.1

Cancel the common factor.

$\frac{9 - 8 w}{5} \cdot 5 < 3 \cdot 5$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.2.1.1.1.2

Rewrite the expression.

$9 - 8 w < 3 \cdot 5$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.2.1.1.2

Reorder $9$ and $- 8 w$ .

$- 8 w + 9 < 3 \cdot 5$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.2.1

Multiply $3$ by $5$ .

$- 8 w + 9 < 15$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3

Solve for $w$ .

Tap for more steps...

Step 1.3.1

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

Tap for more steps...

Step 1.3.1.1

Subtract $9$ from both sides of the inequality.

$- 8 w < 15 - 9$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.1.2

Subtract $9$ from $15$ .

$- 8 w < 6$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2

Divide each term in $- 8 w < 6$ by $- 8$ and simplify.

Tap for more steps...

Step 1.3.2.1

Divide each term in $- 8 w < 6$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 w}{- 8} > \frac{6}{- 8}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.2

Simplify the left side.

Tap for more steps...

Step 1.3.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.3.2.2.1.1

Cancel the common factor.

$\frac{- 8 w}{- 8} > \frac{6}{- 8}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.2.1.2

Divide $w$ by $1$ .

$w > \frac{6}{- 8}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.3.2.3.1

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

Tap for more steps...

Step 1.3.2.3.1.1

Factor $2$ out of $6$ .

$w > \frac{2 (3)}{- 8}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.3.1.2.1

Factor $2$ out of $- 8$ .

$w > \frac{2 \cdot 3}{2 \cdot - 4}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.3.1.2.2

Cancel the common factor.

$w > \frac{2 \cdot 3}{2 \cdot - 4}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.3.1.2.3

Rewrite the expression.

$w > \frac{3}{- 4}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 1.3.2.3.2

Move the negative in front of the fraction.

$w > - \frac{3}{4}$ or $\frac{2}{3} \cdot (18 w - 3) < - 62$

Step 2

Simplify the second inequality.

Tap for more steps...

Step 2.1

Simplify $\frac{2}{3} \cdot (18 w - 3)$ .

Tap for more steps...

Step 2.1.1

Apply the distributive property.

$w > - \frac{3}{4}$ or $\frac{2}{3} \cdot (18 w) + \frac{2}{3} \cdot - 3 < - 62$

Step 2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.2.1

Factor $3$ out of $18 w$ .

$w > - \frac{3}{4}$ or $\frac{2}{3} \cdot (3 (6 w)) + \frac{2}{3} \cdot - 3 < - 62$

Step 2.1.2.2

Cancel the common factor.

$w > - \frac{3}{4}$ or $\frac{2}{3} \cdot (3 (6 w)) + \frac{2}{3} \cdot - 3 < - 62$

Step 2.1.2.3

Rewrite the expression.

$w > - \frac{3}{4}$ or $2 (6 w) + \frac{2}{3} \cdot - 3 < - 62$

Step 2.1.3

Multiply $6$ by $2$ .

$w > - \frac{3}{4}$ or $12 w + \frac{2}{3} \cdot - 3 < - 62$

Step 2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.4.1

Factor $3$ out of $- 3$ .

$w > - \frac{3}{4}$ or $12 w + \frac{2}{3} \cdot (3 (- 1)) < - 62$

Step 2.1.4.2

Cancel the common factor.

$w > - \frac{3}{4}$ or $12 w + \frac{2}{3} \cdot (3 \cdot - 1) < - 62$

Step 2.1.4.3

Rewrite the expression.

$w > - \frac{3}{4}$ or $12 w + 2 \cdot - 1 < - 62$

Step 2.1.5

Multiply $2$ by $- 1$ .

$w > - \frac{3}{4}$ or $12 w - 2 < - 62$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Add $2$ to both sides of the inequality.

$w > - \frac{3}{4}$ or $12 w < - 62 + 2$

Step 2.2.2

Add $- 62$ and $2$ .

$w > - \frac{3}{4}$ or $12 w < - 60$

Step 2.3

Divide each term in $12 w < - 60$ by $12$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $12 w < - 60$ by $12$ .

$w > - \frac{3}{4}$ or $\frac{12 w}{12} < \frac{- 60}{12}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$w > - \frac{3}{4}$ or $\frac{12 w}{12} < \frac{- 60}{12}$

Step 2.3.2.1.2

Divide $w$ by $1$ .

$w > - \frac{3}{4}$ or $w < \frac{- 60}{12}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $- 60$ by $12$ .

$w > - \frac{3}{4}$ or $w < - 5$

Step 3

The union consists of all of the elements that are contained in each interval.

$w < - 5$ or $w > - \frac{3}{4}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$w < - 5 or w > - \frac{3}{4}$

Interval Notation:

$(- \infty, - 5) \cup (- \frac{3}{4}, \infty)$

Step 5