Algebra Examples

$2 x^{2} - 21 x + 4 \geq - 7 x - 8$

Step 1

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

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Step 1.1

Add $7 x$ to both sides of the inequality.

$2 x^{2} - 21 x + 4 + 7 x \geq - 8$

Step 1.2

Add $- 21 x$ and $7 x$ .

$2 x^{2} - 14 x + 4 \geq - 8$

Step 2

Convert the inequality to an equation.

$2 x^{2} - 14 x + 4 = - 8$

Step 3

Add $8$ to both sides of the equation.

$2 x^{2} - 14 x + 4 + 8 = 0$

Step 4

Add $4$ and $8$ .

$2 x^{2} - 14 x + 12 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $2$ out of $2 x^{2} - 14 x + 12$ .

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Step 5.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 14 x + 12 = 0$

Step 5.1.2

Factor $2$ out of $- 14 x$ .

$2 (x^{2}) + 2 (- 7 x) + 12 = 0$

Step 5.1.3

Factor $2$ out of $12$ .

$2 x^{2} + 2 (- 7 x) + 2 \cdot 6 = 0$

Step 5.1.4

Factor $2$ out of $2 x^{2} + 2 (- 7 x)$ .

$2 (x^{2} - 7 x) + 2 \cdot 6 = 0$

Step 5.1.5

Factor $2$ out of $2 (x^{2} - 7 x) + 2 \cdot 6$ .

$2 (x^{2} - 7 x + 6) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor $x^{2} - 7 x + 6$ using the AC method.

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Step 5.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 5.2.1.2

Write the factored form using these integers.

$2 ((x - 6) (x - 1)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$2 (x - 6) (x - 1) = 0$

Step 6

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

$x - 6 = 0$

$x - 1 = 0$

Step 7

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 7.2

Add $6$ to both sides of the equation.

$x = 6$

Step 8

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9

The final solution is all the values that make $2 (x - 6) (x - 1) = 0$ true.

$x = 6, 1$

Step 10

Use each root to create test intervals.

$x < 1$

$1 < x < 6$

$x > 6$

Step 11

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 11.1

Test a value on the interval $x < 1$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 11.1.2

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

$2 {(0)}^{2} - 21 \cdot 0 + 4 \geq - 7 \cdot 0 - 8$

Step 11.1.3

The left side $4$ is greater than the right side $- 8$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $1 < x < 6$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $1 < x < 6$ and see if this value makes the original inequality true.

$x = 4$

Step 11.2.2

Replace $x$ with $4$ in the original inequality.

$2 {(4)}^{2} - 21 \cdot 4 + 4 \geq - 7 \cdot 4 - 8$

Step 11.2.3

The left side $- 48$ is less than the right side $- 36$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $x > 6$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Step 11.3.2

Replace $x$ with $8$ in the original inequality.

$2 {(8)}^{2} - 21 \cdot 8 + 4 \geq - 7 \cdot 8 - 8$

Step 11.3.3

The left side $- 36$ is greater than the right side $- 64$ , which means that the given statement is always true.

True

Step 11.4

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

$x < 1$ True

$1 < x < 6$ False

$x > 6$ True

$x < 1$ True

$1 < x < 6$ False

$x > 6$ True

Step 12

The solution consists of all of the true intervals.

$x \leq 1$ or $x \geq 6$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x \leq 1 or x \geq 6$

Interval Notation:

$(- \infty, 1] \cup [6, \infty)$

Step 14