Calculus Examples

$x^{2} - 10 x \leq - 9$

Step 1

Convert the inequality to an equation.

$x^{2} - 10 x = - 9$

Step 2

Add $9$ to both sides of the equation.

$x^{2} - 10 x + 9 = 0$

Step 3

Factor $x^{2} - 10 x + 9$ using the AC method.

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Step 3.1

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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 3.2

Write the factored form using these integers.

$(x - 9) (x - 1) = 0$

Step 4

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

$x - 9 = 0$

$x - 1 = 0$

Step 5

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

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Step 5.1

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

$x - 9 = 0$

Step 5.2

Add $9$ to both sides of the equation.

$x = 9$

Step 6

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

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Step 6.1

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

$x - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 7

The final solution is all the values that make $(x - 9) (x - 1) = 0$ true.

$x = 9, 1$

Step 8

Use each root to create test intervals.

$x < 1$

$1 < x < 9$

$x > 9$

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 $x < 1$ to see if it makes the inequality true.

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Step 9.1.1

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

$x = 0$

Step 9.1.2

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

${(0)}^{2} - 10 \cdot 0 \leq - 9$

Step 9.1.3

The left side $0$ is greater than the right side $- 9$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $1 < x < 9$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $1 < x < 9$ and see if this value makes the original inequality true.

$x = 5$

Step 9.2.2

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

${(5)}^{2} - 10 \cdot 5 \leq - 9$

Step 9.2.3

The left side $- 25$ is less than the right side $- 9$ , which means that the given statement is always true.

True

Step 9.3

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

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Step 9.3.1

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

$x = 12$

Step 9.3.2

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

${(12)}^{2} - 10 \cdot 12 \leq - 9$

Step 9.3.3

The left side $24$ is greater than the right side $- 9$ , which means that the given statement is false.

False

Step 9.4

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

$x < 1$ False

$1 < x < 9$ True

$x > 9$ False

$x < 1$ False

$1 < x < 9$ True

$x > 9$ False

Step 10

The solution consists of all of the true intervals.

$1 \leq x \leq 9$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$1 \leq x \leq 9$

Interval Notation:

$[1, 9]$

Step 12

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