Solve the Inequality for w 4w^2-9>=0

$4 w^{2} - 9 \geq 0$

Step 1

Add $9$ to both sides of the inequality.

$4 w^{2} \geq 9$

Step 2

Divide each term in $4 w^{2} \geq 9$ by $4$ and simplify.

Divide each term in $4 w^{2} \geq 9$ by $4$ .

$\frac{4 w^{2}}{4} \geq \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 w^{2}}{4} \geq \frac{9}{4}$

Divide $w^{2}$ by $1$ .

$w^{2} \geq \frac{9}{4}$

Step 3

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{w^{2}} \geq \sqrt{\frac{9}{4}}$

Step 4

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| w | \geq \sqrt{\frac{9}{4}}$

Simplify the right side.

Simplify $\sqrt{\frac{9}{4}}$ .

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$| w | \geq \frac{\sqrt{9}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$| w | \geq \frac{\sqrt{3^{2}}}{\sqrt{4}}$

Pull terms out from under the radical.

$| w | \geq \frac{| 3 |}{\sqrt{4}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| w | \geq \frac{3}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$| w | \geq \frac{3}{\sqrt{2^{2}}}$

Pull terms out from under the radical.

$| w | \geq \frac{3}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| w | \geq \frac{3}{2}$

Step 5

Write $| w | \geq \frac{3}{2}$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$w \geq 0$

In the piece where $w$ is non-negative, remove the absolute value.

$w \geq \frac{3}{2}$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$w < 0$

In the piece where $w$ is negative, remove the absolute value and multiply by $- 1$ .

$- w \geq \frac{3}{2}$

Write as a piecewise.

${\begin{cases} w \geq \frac{3}{2} & w \geq 0 \\ - w \geq \frac{3}{2} & w < 0 \end{cases}$

Step 6

Find the intersection of $w \geq \frac{3}{2}$ and $w \geq 0$ .

$w \geq \frac{3}{2}$

Step 7

Divide each term in $- w \geq \frac{3}{2}$ by $- 1$ and simplify.

Divide each term in $- w \geq \frac{3}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- w}{- 1} \leq \frac{\frac{3}{2}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{w}{1} \leq \frac{\frac{3}{2}}{- 1}$

Divide $w$ by $1$ .

$w \leq \frac{\frac{3}{2}}{- 1}$

Simplify the right side.

Move the negative one from the denominator of $\frac{\frac{3}{2}}{- 1}$ .

$w \leq - 1 \cdot \frac{3}{2}$

Rewrite $- 1 \cdot \frac{3}{2}$ as $- \frac{3}{2}$ .

$w \leq - \frac{3}{2}$

Step 8

Find the union of the solutions.

$w \leq - \frac{3}{2}$ or $w \geq \frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$w \leq - \frac{3}{2} or w \geq \frac{3}{2}$

Interval Notation:

$(- \infty, - \frac{3}{2}] \cup [\frac{3}{2}, \infty)$

Step 10

Solve the Inequality for w 4w^2-9>=0

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