Algebra Examples

$4 x^{2} - 42 - 102 < 0$

Step 1

Subtract $102$ from $- 42$ .

$4 x^{2} - 144 < 0$

Step 2

Add $144$ to both sides of the inequality.

$4 x^{2} < 144$

Step 3

Divide each term in $4 x^{2} < 144$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x^{2} < 144$ by $4$ .

$\frac{4 x^{2}}{4} < \frac{144}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} < \frac{144}{4}$

Step 3.2.1.2

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

$x^{2} < \frac{144}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $144$ by $4$ .

$x^{2} < 36$

Step 4

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

$\sqrt{x^{2}} < \sqrt{36}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Pull terms out from under the radical.

$| x | < \sqrt{36}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\sqrt{36}$ .

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Step 5.2.1.1

Rewrite $36$ as $6^{2}$ .

$| x | < \sqrt{6^{2}}$

Step 5.2.1.2

Pull terms out from under the radical.

$| x | < | 6 |$

Step 5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$| x | < 6$

Step 6

Write $| x | < 6$ as a piecewise.

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Step 6.1

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

$x \geq 0$

Step 6.2

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

$x < 6$

Step 6.3

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

$x < 0$

Step 6.4

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

$- x < 6$

Step 6.5

Write as a piecewise.

${\begin{cases} x < 6 & x \geq 0 \\ - x < 6 & x < 0 \end{cases}$

Step 7

Find the intersection of $x < 6$ and $x \geq 0$ .

$0 \leq x < 6$

Step 8

Solve $- x < 6$ when $x < 0$ .

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Step 8.1

Divide each term in $- x < 6$ by $- 1$ and simplify.

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Step 8.1.1

Divide each term in $- x < 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{6}{- 1}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{6}{- 1}$

Step 8.1.2.2

Divide $x$ by $1$ .

$x > \frac{6}{- 1}$

Step 8.1.3

Simplify the right side.

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Step 8.1.3.1

Divide $6$ by $- 1$ .

$x > - 6$

Step 8.2

Find the intersection of $x > - 6$ and $x < 0$ .

$- 6 < x < 0$

Step 9

Find the union of the solutions.

$- 6 < x < 6$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < 6$

Interval Notation:

$(- 6, 6)$

Step 11