Solve for x x^2-10x<-24

$x^{2} - 10 x < - 24$

Step 1

Convert the inequality to an equation.

$x^{2} - 10 x = - 24$

Step 2

Add $24$ to both sides of the equation.

$x^{2} - 10 x + 24 = 0$

Step 3

Factor $x^{2} - 10 x + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Write the factored form using these integers.

$(x - 6) (x - 4) = 0$

Step 4

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

$x - 6 = 0$

$x - 4 = 0$

Step 5

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 6

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 7

The final solution is all the values that make $(x - 6) (x - 4) = 0$ true.

$x = 6, 4$

Step 8

Use each root to create test intervals.

$x < 4$

$4 < x < 6$

$x > 6$

Step 9

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 $x < 4$ to see if it makes the inequality true.

Choose a value on the interval $x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} - 10 \cdot 0 < - 24$

The left side $0$ is not less than the right side $- 24$ , which means that the given statement is false.

False

Test a value on the interval $4 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 6$ and see if this value makes the original inequality true.

$x = 5$

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

${(5)}^{2} - 10 \cdot 5 < - 24$

The left side $- 25$ is less than the right side $- 24$ , which means that the given statement is always true.

True

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

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

$x = 8$

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

${(8)}^{2} - 10 \cdot 8 < - 24$

The left side $- 16$ is not less than the right side $- 24$ , which means that the given statement is false.

False

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

$x < 4$ False

$4 < x < 6$ True

$x > 6$ False

$x < 4$ False

$4 < x < 6$ True

$x > 6$ False

Step 10

The solution consists of all of the true intervals.

$4 < x < 6$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$4 < x < 6$

Interval Notation:

$(4, 6)$

Step 12

Solve for x x^2-10x<-24

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