Algebra Examples

$\log (7 x - 11) \leq 2$

Step 1

Convert the inequality to an equality.

$\log (7 x - 11) = 2$

Step 2

Solve the equation.

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Step 2.1

Rewrite $\log (7 x - 11) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{2} = 7 x - 11$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $7 x - 11 = 10^{2}$ .

$7 x - 11 = 10^{2}$

Step 2.2.2

Raise $10$ to the power of $2$ .

$7 x - 11 = 100$

Step 2.2.3

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

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Step 2.2.3.1

Add $11$ to both sides of the equation.

$7 x = 100 + 11$

Step 2.2.3.2

Add $100$ and $11$ .

$7 x = 111$

Step 2.2.4

Divide each term in $7 x = 111$ by $7$ and simplify.

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Step 2.2.4.1

Divide each term in $7 x = 111$ by $7$ .

$\frac{7 x}{7} = \frac{111}{7}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $7$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{111}{7}$

Step 2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{111}{7}$

Step 3

Find the domain of $\log (7 x - 11) - 2$ .

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Step 3.1

Set the argument in $\log (7 x - 11)$ greater than $0$ to find where the expression is defined.

$7 x - 11 > 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Add $11$ to both sides of the inequality.

$7 x > 11$

Step 3.2.2

Divide each term in $7 x > 11$ by $7$ and simplify.

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Step 3.2.2.1

Divide each term in $7 x > 11$ by $7$ .

$\frac{7 x}{7} > \frac{11}{7}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $7$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} > \frac{11}{7}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{11}{7}$

Step 3.3

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

$(\frac{11}{7}, \infty)$

Step 4

Use each root to create test intervals.

$x < \frac{11}{7}$

$\frac{11}{7} < x < \frac{111}{7}$

$x > \frac{111}{7}$

Step 5

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 5.1

Test a value on the interval $x < \frac{11}{7}$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < \frac{11}{7}$ and see if this value makes the original inequality true.

$x = 0$

Step 5.1.2

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

$\log (7 (0) - 11) \leq 2$

Step 5.1.3

Determine if the inequality is true.

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Step 5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.1.3.2

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

False

Step 5.2

Test a value on the interval $\frac{11}{7} < x < \frac{111}{7}$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $\frac{11}{7} < x < \frac{111}{7}$ and see if this value makes the original inequality true.

$x = 4$

Step 5.2.2

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

$\log (7 (4) - 11) \leq 2$

Step 5.2.3

The left side $1.23044892$ is less than the right side $2$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > \frac{111}{7}$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > \frac{111}{7}$ and see if this value makes the original inequality true.

$x = 18$

Step 5.3.2

Replace $x$ with $18$ in the original inequality.

$\log (7 (18) - 11) \leq 2$

Step 5.3.3

The left side $2.06069784$ is greater than the right side $2$ , which means that the given statement is false.

False

Step 5.4

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

$x < \frac{11}{7}$ False

$\frac{11}{7} < x < \frac{111}{7}$ True

$x > \frac{111}{7}$ False

$x < \frac{11}{7}$ False

$\frac{11}{7} < x < \frac{111}{7}$ True

$x > \frac{111}{7}$ False

Step 6

The solution consists of all of the true intervals.

$\frac{11}{7} < x \leq \frac{111}{7}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$\frac{11}{7} < x \leq \frac{111}{7}$

Interval Notation:

$(\frac{11}{7}, \frac{111}{7}]$

Step 8