Solve the Inequality for x x^2-10x+21<=0

$x^{2} - 10 x + 21 \leq 0$

Step 1

Convert the inequality to an equation.

$x^{2} - 10 x + 21 = 0$

Step 2

Factor $x^{2} - 10 x + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

$(x - 7) (x - 3) = 0$

Step 3

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

$x - 7 = 0$

$x - 3 = 0$

Step 4

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 5

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 6

The final solution is all the values that make $(x - 7) (x - 3) = 0$ true.

$x = 7, 3$

Step 7

Use each root to create test intervals.

$x < 3$

$3 < x < 7$

$x > 7$

Step 8

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 $x < 3$ to see if it makes the inequality true.

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

$x = 0$

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

${(0)}^{2} - 10 \cdot 0 + 21 \leq 0$

The left side $21$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $3 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 7$ and see if this value makes the original inequality true.

$x = 5$

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

${(5)}^{2} - 10 \cdot 5 + 21 \leq 0$

The left side $- 4$ is less than the right side $0$ , 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.

${(10)}^{2} - 10 \cdot 10 + 21 \leq 0$

The left side $21$ is greater than the right side $0$ , which means that the given statement is false.

False

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

$x < 3$ False

$3 < x < 7$ True

$x > 7$ False

$x < 3$ False

$3 < x < 7$ True

$x > 7$ False

Step 9

The solution consists of all of the true intervals.

$3 \leq x \leq 7$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$3 \leq x \leq 7$

Interval Notation:

$[3, 7]$

Step 11

Solve the Inequality for x x^2-10x+21<=0

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