Algebra Examples

$2 x^{2} + x \geq 15$

Step 1

Convert the inequality to an equation.

$2 x^{2} + x = 15$

Step 2

Subtract $15$ from both sides of the equation.

$2 x^{2} + x - 15 = 0$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = 1$ .

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Step 3.1.1

Multiply by $1$ .

$2 x^{2} + 1 x - 15 = 0$

Step 3.1.2

Rewrite $1$ as $- 5$ plus $6$

$2 x^{2} + (- 5 + 6) x - 15 = 0$

Step 3.1.3

Apply the distributive property.

$2 x^{2} - 5 x + 6 x - 15 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(2 x^{2} - 5 x) + 6 x - 15 = 0$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x - 5) + 3 (2 x - 5) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 5$ .

$(2 x - 5) (x + 3) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 5 = 0$

$x + 3 = 0$

Step 5

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 5.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 5.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 5.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 6

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

The final solution is all the values that make $(2 x - 5) (x + 3) = 0$ true.

$x = \frac{5}{2}, - 3$

Step 8

Use each root to create test intervals.

$x < - 3$

$- 3 < x < \frac{5}{2}$

$x > \frac{5}{2}$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 9.1

Test a value on the interval $x < - 3$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

Step 9.1.2

Replace $x$ with $- 6$ in the original inequality.

$2 {(- 6)}^{2} - 6 \geq 15$

Step 9.1.3

The left side $66$ is greater than the right side $15$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $- 3 < x < \frac{5}{2}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 3 < x < \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

Replace $x$ with $0$ in the original inequality.

$2 {(0)}^{2} + 0 \geq 15$

Step 9.2.3

The left side $0$ is less than the right side $15$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > \frac{5}{2}$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 5$

Step 9.3.2

Replace $x$ with $5$ in the original inequality.

$2 {(5)}^{2} + 5 \geq 15$

Step 9.3.3

The left side $55$ is greater than the right side $15$ , which means that the given statement is always true.

True

Step 9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 3$ True

$- 3 < x < \frac{5}{2}$ False

$x > \frac{5}{2}$ True

$x < - 3$ True

$- 3 < x < \frac{5}{2}$ False

$x > \frac{5}{2}$ True

Step 10

The solution consists of all of the true intervals.

$x \leq - 3$ or $x \geq \frac{5}{2}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3 or x \geq \frac{5}{2}$

Interval Notation:

$(- \infty, - 3] \cup [\frac{5}{2}, \infty)$

Step 12