Solve the Inequality for x 2 log of x< log of 2-x

$2 \log (x) < \log (2 - x)$

Step 1

Convert the inequality to an equality.

$2 \log (x) = \log (2 - x)$

Step 2

Solve the equation.

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) = \log (2 - x)$

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} = 2 - x$

Solve for $x$ .

Add $x$ to both sides of the equation.

$x^{2} + x = 2$

Subtract $2$ from both sides of the equation.

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

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.

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

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

$x - 1 = 0$

$x + 2 = 0$

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.

$x = 1$

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.

$x = - 2$

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

$x = 1, - 2$

Step 3

Find the domain of $\log (\frac{x^{2}}{2 - x})$ .

Set the argument in $\log (\frac{x^{2}}{2 - x})$ greater than $0$ to find where the expression is defined.

$\frac{x^{2}}{2 - x} > 0$

Solve for $x$ .

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

$x^{2} = 0$

$2 - x = 0$

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

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

$x = 0$

$x = 2$

Consolidate the solutions.

$x = 0, 2$

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

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

$2 - x = 0$

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

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

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

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

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 $x < 0$ to see if it makes the inequality true.

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$\frac{{(- 2)}^{2}}{2 - (- 2)} > 0$

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Replace $x$ with $1$ in the original inequality.

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

The left side $1$ is greater 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.

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

The left side $- 8$ is not 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.

$x < 0$ True

$0 < x < 2$ True

$x > 2$ False

$x < 0$ True

$0 < x < 2$ True

$x > 2$ False

The solution consists of all of the true intervals.

$x < 0$ or $0 < x < 2$

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

$2 - x = 0$

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

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

$(- \infty, 0) \cup (0, 2)$

Step 4

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 1$

$x > 1$

Step 5

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.

$2 \log (- 4) < \log (2 - (- 4))$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

The left side has no solution, 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.

$2 \log (0) < \log (2 - (0))$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

The left side has no solution, which means that the given statement is false.

False

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.

$2 \log (4) < \log (2 - (4))$

Determine if the inequality is true.

The equation cannot be solved because it is undefined.

$Undefined$

The right side has no solution, which means that the given statement is false.

False

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

$x < - 2$ False

$- 2 < x < 1$ False

$x > 1$ False

$x < - 2$ False

$- 2 < x < 1$ False

$x > 1$ False

Step 6

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Solve the Inequality for x 2 log of x< log of 2-x

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