Solve the Inequality for x x^2-53<=-2x+10

$x^{2} - 53 \leq - 2 x + 10$

Step 1

Add $2 x$ to both sides of the inequality.

$x^{2} - 53 + 2 x \leq 10$

Step 2

Convert the inequality to an equation.

$x^{2} - 53 + 2 x = 10$

Step 3

Subtract $10$ from both sides of the equation.

$x^{2} - 53 + 2 x - 10 = 0$

Step 4

Subtract $10$ from $- 53$ .

$x^{2} + 2 x - 63 = 0$

Step 5

Factor $x^{2} + 2 x - 63$ using the AC method.

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 $- 63$ and whose sum is $2$ .

$- 7, 9$

Write the factored form using these integers.

$(x - 7) (x + 9) = 0$

Step 6

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

$x - 7 = 0$

$x + 9 = 0$

Step 7

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 8

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 9

The final solution is all the values that make $(x - 7) (x + 9) = 0$ true.

$x = 7, - 9$

Step 10

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 7$

$x > 7$

Step 11

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

Test a value on the interval $x < - 9$ to see if it makes the inequality true.

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Replace $x$ with $- 12$ in the original inequality.

${(- 12)}^{2} - 53 \leq - 2 \cdot - 12 + 10$

The left side $91$ is greater than the right side $34$ , which means that the given statement is false.

False

Test a value on the interval $- 9 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} - 53 \leq - 2 \cdot 0 + 10$

The left side $- 53$ is less than the right side $10$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

${(10)}^{2} - 53 \leq - 2 \cdot 10 + 10$

The left side $47$ is greater than the right side $- 10$ , which means that the given statement is false.

False

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

$x < - 9$ False

$- 9 < x < 7$ True

$x > 7$ False

$x < - 9$ False

$- 9 < x < 7$ True

$x > 7$ False

Step 12

The solution consists of all of the true intervals.

$- 9 \leq x \leq 7$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 9 \leq x \leq 7$

Interval Notation:

$[- 9, 7]$

Step 14

Solve the Inequality for x x^2-53<=-2x+10

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