Solve the Inequality for x x^2+x+7<10x-1

$x^{2} + x + 7 < 10 x - 1$

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

Subtract $10 x$ from both sides of the inequality.

$x^{2} + x + 7 - 10 x < - 1$

Subtract $10 x$ from $x$ .

$x^{2} - 9 x + 7 < - 1$

Convert the inequality to an equation.

$x^{2} - 9 x + 7 = - 1$

Add $1$ to both sides of the equation.

$x^{2} - 9 x + 7 + 1 = 0$

Add $7$ and $1$ .

$x^{2} - 9 x + 8 = 0$

Factor $x^{2} - 9 x + 8$ 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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Write the factored form using these integers.

$(x - 8) (x - 1) = 0$

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

$x - 8 = 0$

$x - 1 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 8) (x - 1) = 0$ true.

$x = 8, 1$

Use each root to create test intervals.

$x < 1$

$1 < x < 8$

$x > 8$

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

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

$x = 0$

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

${(0)}^{2} + 0 + 7 < 10 (0) - 1$

The left side $7$ is not less than the right side $- 1$ , which means that the given statement is false.

False

Test a value on the interval $1 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 8$ and see if this value makes the original inequality true.

$x = 4$

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

${(4)}^{2} + 4 + 7 < 10 (4) - 1$

The left side $27$ is less than the right side $39$ , which means that the given statement is always true.

True

Test a value on the interval $x > 8$ to see if it makes the inequality true.

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

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

${(10)}^{2} + 10 + 7 < 10 (10) - 1$

The left side $117$ is not less than the right side $99$ , which means that the given statement is false.

False

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

$x < 1$ False

$1 < x < 8$ True

$x > 8$ False

$x < 1$ False

$1 < x < 8$ True

$x > 8$ False

The solution consists of all of the true intervals.

$1 < x < 8$

The result can be shown in multiple forms.

Inequality Form:

$1 < x < 8$

Interval Notation:

$(1, 8)$

Solve the Inequality for x x^2+x+7<10x-1

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