Solve the Inequality for x (x^2-1)/(x^2-4)<=0

$\frac{x^{2} - 1}{x^{2} - 4} \leq 0$

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} - 1 = 0$

$x^{2} - 4 = 0$

Add $1$ to both sides of the equation.

$x^{2} = 1$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Add $4$ to both sides of the equation.

$x^{2} = 4$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 1, - 1$

$x = 2, - 2$

Consolidate the solutions.

$x = 1, - 1, 2, - 2$

Find the domain of $\frac{x^{2} - 1}{x^{2} - 4}$ .

Set the denominator in $\frac{x^{2} - 1}{x^{2} - 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 = 0$

Solve for $x$ .

Add $4$ to both sides of the equation.

$x^{2} = 4$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Use each root to create test intervals.

$x < - 2$

$- 2 < x < - 1$

$- 1 < x < 1$

$1 < x < 2$

$x > 2$

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.

$\frac{{(- 4)}^{2} - 1}{{(- 4)}^{2} - 4} \leq 0$

The left side $1.25$ 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.

$x = - 1.5$

Replace $x$ with $- 1.5$ in the original inequality.

$\frac{{(- 1.5)}^{2} - 1}{{(- 1.5)}^{2} - 4} \leq 0$

The left side $- 0.\overline{714285}$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

$\frac{{(0)}^{2} - 1}{{(0)}^{2} - 4} \leq 0$

The left side $0.25$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 1.5$

Replace $x$ with $1.5$ in the original inequality.

$\frac{{(1.5)}^{2} - 1}{{(1.5)}^{2} - 4} \leq 0$

The left side $- 0.\overline{714285}$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$\frac{{(4)}^{2} - 1}{{(4)}^{2} - 4} \leq 0$

The left side $1.25$ 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

$- 1 < x < 1$ False

$1 < x < 2$ True

$x > 2$ False

$x < - 2$ False

$- 2 < x < - 1$ True

$- 1 < x < 1$ False

$1 < x < 2$ True

$x > 2$ False

The solution consists of all of the true intervals.

$- 2 < x \leq - 1$ or $1 \leq x < 2$

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x \leq - 1 or 1 \leq x < 2$

Interval Notation:

$(- 2, - 1] \cup [1, 2)$

Solve the Inequality for x (x^2-1)/(x^2-4)<=0

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