Solve the Inequality for w 1-7/6w<=9/4-5/6w

$1 - \frac{7}{6} w \leq \frac{9}{4} - \frac{5}{6} w$

Step 1

Simplify each term.

Combine $w$ and $\frac{7}{6}$ .

$1 - \frac{w \cdot 7}{6} \leq \frac{9}{4} - \frac{5}{6} w$

Move $7$ to the left of $w$ .

$1 - \frac{7 w}{6} \leq \frac{9}{4} - \frac{5}{6} w$

Step 2

Simplify each term.

Combine $w$ and $\frac{5}{6}$ .

$1 - \frac{7 w}{6} \leq \frac{9}{4} - \frac{w \cdot 5}{6}$

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

$1 - \frac{7 w}{6} \leq \frac{9}{4} - \frac{5 w}{6}$

Step 3

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

Add $\frac{5 w}{6}$ to both sides of the inequality.

$1 - \frac{7 w}{6} + \frac{5 w}{6} \leq \frac{9}{4}$

Combine the numerators over the common denominator.

$1 + \frac{- 7 w + 5 w}{6} \leq \frac{9}{4}$

Add $- 7 w$ and $5 w$ .

$1 + \frac{- 2 w}{6} \leq \frac{9}{4}$

Simplify each term.

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2 w$ .

$1 + \frac{2 (- w)}{6} \leq \frac{9}{4}$

Cancel the common factors.

Factor $2$ out of $6$ .

$1 + \frac{2 (- w)}{2 (3)} \leq \frac{9}{4}$

Cancel the common factor.

$1 + \frac{2 (- w)}{2 \cdot 3} \leq \frac{9}{4}$

Rewrite the expression.

$1 + \frac{- w}{3} \leq \frac{9}{4}$

Move the negative in front of the fraction.

$1 - \frac{w}{3} \leq \frac{9}{4}$

Step 4

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

Subtract $1$ from both sides of the inequality.

$- \frac{w}{3} \leq \frac{9}{4} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{w}{3} \leq \frac{9}{4} - 1 \cdot \frac{4}{4}$

Combine $- 1$ and $\frac{4}{4}$ .

$- \frac{w}{3} \leq \frac{9}{4} + \frac{- 1 \cdot 4}{4}$

Combine the numerators over the common denominator.

$- \frac{w}{3} \leq \frac{9 - 1 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 1$ by $4$ .

$- \frac{w}{3} \leq \frac{9 - 4}{4}$

Subtract $4$ from $9$ .

$- \frac{w}{3} \leq \frac{5}{4}$

Step 5

Divide each term in $- \frac{w}{3} \leq \frac{5}{4}$ by $- 1$ and simplify.

Divide each term in $- \frac{w}{3} \leq \frac{5}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{w}{3}}{- 1} \geq \frac{\frac{5}{4}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{w}{3}}{1} \geq \frac{\frac{5}{4}}{- 1}$

Divide $\frac{w}{3}$ by $1$ .

$\frac{w}{3} \geq \frac{\frac{5}{4}}{- 1}$

Simplify the right side.

Move the negative one from the denominator of $\frac{\frac{5}{4}}{- 1}$ .

$\frac{w}{3} \geq - 1 \cdot \frac{5}{4}$

Rewrite $- 1 \cdot \frac{5}{4}$ as $- \frac{5}{4}$ .

$\frac{w}{3} \geq - \frac{5}{4}$

Step 6

Multiply both sides by $3$ .

$\frac{w}{3} \cdot 3 \geq - \frac{5}{4} \cdot 3$

Step 7

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{w}{3} \cdot 3 \geq - \frac{5}{4} \cdot 3$

Rewrite the expression.

$w \geq - \frac{5}{4} \cdot 3$

Simplify the right side.

Simplify $- \frac{5}{4} \cdot 3$ .

Multiply $- \frac{5}{4} \cdot 3$ .

Multiply $3$ by $- 1$ .

$w \geq - 3 (\frac{5}{4})$

Combine $- 3$ and $\frac{5}{4}$ .

$w \geq \frac{- 3 \cdot 5}{4}$

Multiply $- 3$ by $5$ .

$w \geq \frac{- 15}{4}$

Move the negative in front of the fraction.

$w \geq - \frac{15}{4}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$w \geq - \frac{15}{4}$

Interval Notation:

$[- \frac{15}{4}, \infty)$

Solve the Inequality for w 1-7/6w<=9/4-5/6w

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