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

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

Step 1

Convert the inequality to an equation.

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

Step 2

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$(x - 1) (x + 2) = 0$

Step 3

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

$x - 1 = 0$

$x + 2 = 0$

Step 4

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.

$x = 1$

Step 5

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

The final solution is all the values that make $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

Step 7

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 1$

$x > 1$

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 $x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

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

${(- 4)}^{2} - 4 - 2 \leq 0$

The left side $10$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

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

The left side $- 2$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$x = 4$

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

${(4)}^{2} + 4 - 2 \leq 0$

The left side $18$ 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.

$x < - 2$ False

$- 2 < x < 1$ True

$x > 1$ False

$x < - 2$ False

$- 2 < x < 1$ True

$x > 1$ False

Step 9

The solution consists of all of the true intervals.

$- 2 \leq x \leq 1$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x \leq 1$

Interval Notation:

$[- 2, 1]$

Step 11

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

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