Solve the Inequality for x log base 4 of 2x+5< = log base 4 of 4x-3

$\log_{4} (2 x + 5) \leq \log_{4} (4 x - 3)$

Step 1

Convert the inequality to an equality.

$\log_{4} (2 x + 5) = \log_{4} (4 x - 3)$

Step 2

Solve the equation.

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

$2 x + 5 = 4 x - 3$

Solve for $x$ .

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

Subtract $4 x$ from both sides of the equation.

$2 x + 5 - 4 x = - 3$

Subtract $4 x$ from $2 x$ .

$- 2 x + 5 = - 3$

Move all terms not containing $x$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$- 2 x = - 3 - 5$

Subtract $5$ from $- 3$ .

$- 2 x = - 8$

Divide each term in $- 2 x = - 8$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 8$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 8}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 8}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 8}{- 2}$

Simplify the right side.

Divide $- 8$ by $- 2$ .

$x = 4$

Step 3

Find the domain of $\log_{4} (\frac{2 x + 5}{4 x - 3})$ .

Set the argument in $\log_{4} (\frac{2 x + 5}{4 x - 3})$ greater than $0$ to find where the expression is defined.

$\frac{2 x + 5}{4 x - 3} > 0$

Solve for $x$ .

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

$2 x + 5 = 0$

$4 x - 3 = 0$

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Add $3$ to both sides of the equation.

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{3}{4}$

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

$x = - \frac{5}{2}$

$x = \frac{3}{4}$

Consolidate the solutions.

$x = - \frac{5}{2}, \frac{3}{4}$

Find the domain of $\frac{2 x + 5}{4 x - 3}$ .

Set the denominator in $\frac{2 x + 5}{4 x - 3}$ equal to $0$ to find where the expression is undefined.

$4 x - 3 = 0$

Solve for $x$ .

Add $3$ to both sides of the equation.

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{3}{4}$

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

$(- \infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty)$

Use each root to create test intervals.

$x < - \frac{5}{2}$

$- \frac{5}{2} < x < \frac{3}{4}$

$x > \frac{3}{4}$

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 $x < - \frac{5}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{5}{2}$ and see if this value makes the original inequality true.

$x = - 5$

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

$\frac{2 (- 5) + 5}{4 (- 5) - 3} > 0$

The left side $0.2173913$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- \frac{5}{2} < x < \frac{3}{4}$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{5}{2} < x < \frac{3}{4}$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{2 (0) + 5}{4 (0) - 3} > 0$

The left side $- 1.\overline{6}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{3}{4}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{3}{4}$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$\frac{2 (3) + 5}{4 (3) - 3} > 0$

The left side $1.\overline{2}$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - \frac{5}{2}$ True

$- \frac{5}{2} < x < \frac{3}{4}$ False

$x > \frac{3}{4}$ True

$x < - \frac{5}{2}$ True

$- \frac{5}{2} < x < \frac{3}{4}$ False

$x > \frac{3}{4}$ True

The solution consists of all of the true intervals.

$x < - \frac{5}{2}$ or $x > \frac{3}{4}$

Set the denominator in $\frac{2 x + 5}{4 x - 3}$ equal to $0$ to find where the expression is undefined.

$4 x - 3 = 0$

Solve for $x$ .

Add $3$ to both sides of the equation.

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{3}{4}$

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

$(- \infty, - \frac{5}{2}) \cup (\frac{3}{4}, \infty)$

Step 4

The solution consists of all of the true intervals.

$x \geq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \geq 4$

Interval Notation:

$[4, \infty)$

Step 6

Solve the Inequality for x log base 4 of 2x+5< = log base 4 of 4x-3

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