Algebra Examples

$4 x^{2} - 3 x - 11 \geq - 3 x - 10$

Step 1

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

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Step 1.1

Add $3 x$ to both sides of the inequality.

$4 x^{2} - 3 x - 11 + 3 x \geq - 10$

Step 1.2

Combine the opposite terms in $4 x^{2} - 3 x - 11 + 3 x$ .

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Step 1.2.1

Add $- 3 x$ and $3 x$ .

$4 x^{2} + 0 - 11 \geq - 10$

Step 1.2.2

Add $4 x^{2}$ and $0$ .

$4 x^{2} - 11 \geq - 10$

Step 2

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

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Step 2.1

Add $11$ to both sides of the inequality.

$4 x^{2} \geq - 10 + 11$

Step 2.2

Add $- 10$ and $11$ .

$4 x^{2} \geq 1$

Step 3

Divide each term in $4 x^{2} \geq 1$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x^{2} \geq 1$ by $4$ .

$\frac{4 x^{2}}{4} \geq \frac{1}{4}$

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.

$\frac{4 x^{2}}{4} \geq \frac{1}{4}$

Step 3.2.1.2

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

$x^{2} \geq \frac{1}{4}$

Step 4

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

$\sqrt{x^{2}} \geq \sqrt{\frac{1}{4}}$

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.

$| x | \geq \sqrt{\frac{1}{4}}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\sqrt{\frac{1}{4}}$ .

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Step 5.2.1.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$| x | \geq \frac{\sqrt{1}}{\sqrt{4}}$

Step 5.2.1.2

Any root of $1$ is $1$ .

$| x | \geq \frac{1}{\sqrt{4}}$

Step 5.2.1.3

Simplify the denominator.

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Step 5.2.1.3.1

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

$| x | \geq \frac{1}{\sqrt{2^{2}}}$

Step 5.2.1.3.2

Pull terms out from under the radical.

$| x | \geq \frac{1}{| 2 |}$

Step 5.2.1.3.3

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

$| x | \geq \frac{1}{2}$

Step 6

Write $| x | \geq \frac{1}{2}$ 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.

$x \geq \frac{1}{2}$

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 $- 1$ .

$- x \geq \frac{1}{2}$

Step 6.5

Write as a piecewise.

${\begin{cases} x \geq \frac{1}{2} & x \geq 0 \\ - x \geq \frac{1}{2} & x < 0 \end{cases}$

Step 7

Find the intersection of $x \geq \frac{1}{2}$ and $x \geq 0$ .

$x \geq \frac{1}{2}$

Step 8

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

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Step 8.1

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

$\frac{- x}{- 1} \leq \frac{\frac{1}{2}}{- 1}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{\frac{1}{2}}{- 1}$

Step 8.2.2

Divide $x$ by $1$ .

$x \leq \frac{\frac{1}{2}}{- 1}$

Step 8.3

Simplify the right side.

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Step 8.3.1

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

$x \leq - 1 \cdot \frac{1}{2}$

Step 8.3.2

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

$x \leq - \frac{1}{2}$

Step 9

Find the union of the solutions.

$x \leq - \frac{1}{2}$ or $x \geq \frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{1}{2} or x \geq \frac{1}{2}$

Interval Notation:

$(- \infty, - \frac{1}{2}] \cup [\frac{1}{2}, \infty)$

Step 11