Precalculus Examples

$3 + \sqrt{4 a - 5} \leq 10$

Step 1

Move all terms not containing $\sqrt{4 a - 5}$ to the right side of the inequality.

Tap for more steps...

Step 1.1

Subtract $3$ from both sides of the inequality.

$\sqrt{4 a - 5} \leq 10 - 3$

Step 1.2

Subtract $3$ from $10$ .

$\sqrt{4 a - 5} \leq 7$

Step 2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{4 a - 5}}^{2} \leq 7^{2}$

Step 3

Simplify each side of the inequality.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 a - 5}$ as ${(4 a - 5)}^{\frac{1}{2}}$ .

${({(4 a - 5)}^{\frac{1}{2}})}^{2} \leq 7^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${({(4 a - 5)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${({(4 a - 5)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 a - 5)}^{\frac{1}{2} \cdot 2} \leq 7^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

${(4 a - 5)}^{\frac{1}{2} \cdot 2} \leq 7^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 a - 5)}^{1} \leq 7^{2}$

Step 3.2.1.2

Simplify.

$4 a - 5 \leq 7^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Raise $7$ to the power of $2$ .

$4 a - 5 \leq 49$

Step 4

Solve for $a$ .

Tap for more steps...

Step 4.1

Move all terms not containing $a$ to the right side of the inequality.

Tap for more steps...

Step 4.1.1

Add $5$ to both sides of the inequality.

$4 a \leq 49 + 5$

Step 4.1.2

Add $49$ and $5$ .

$4 a \leq 54$

Step 4.2

Divide each term in $4 a \leq 54$ by $4$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $4 a \leq 54$ by $4$ .

$\frac{4 a}{4} \leq \frac{54}{4}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 a}{4} \leq \frac{54}{4}$

Step 4.2.2.1.2

Divide $a$ by $1$ .

$a \leq \frac{54}{4}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Cancel the common factor of $54$ and $4$ .

Tap for more steps...

Step 4.2.3.1.1

Factor $2$ out of $54$ .

$a \leq \frac{2 (27)}{4}$

Step 4.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.1.2.1

Factor $2$ out of $4$ .

$a \leq \frac{2 \cdot 27}{2 \cdot 2}$

Step 4.2.3.1.2.2

Cancel the common factor.

$a \leq \frac{2 \cdot 27}{2 \cdot 2}$

Step 4.2.3.1.2.3

Rewrite the expression.

$a \leq \frac{27}{2}$

Step 5

Find the domain of $- 7 + \sqrt{4 a - 5}$ .

Tap for more steps...

Step 5.1

Set the radicand in $\sqrt{4 a - 5}$ greater than or equal to $0$ to find where the expression is defined.

$4 a - 5 \geq 0$

Step 5.2

Solve for $a$ .

Tap for more steps...

Step 5.2.1

Add $5$ to both sides of the inequality.

$4 a \geq 5$

Step 5.2.2

Divide each term in $4 a \geq 5$ by $4$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in $4 a \geq 5$ by $4$ .

$\frac{4 a}{4} \geq \frac{5}{4}$

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{4 a}{4} \geq \frac{5}{4}$

Step 5.2.2.2.1.2

Divide $a$ by $1$ .

$a \geq \frac{5}{4}$

Step 5.3

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

$[\frac{5}{4}, \infty)$

Step 6

Use each root to create test intervals.

$a < \frac{5}{4}$

$\frac{5}{4} < a < \frac{27}{2}$

$a > \frac{27}{2}$

Step 7

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

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

$a = 0$

Step 7.1.2

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

$3 + \sqrt{4 (0) - 5} \leq 10$

Step 7.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 7.2

Test a value on the interval $\frac{5}{4} < a < \frac{27}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 7.2.1

Choose a value on the interval $\frac{5}{4} < a < \frac{27}{2}$ and see if this value makes the original inequality true.

$a = 4$

Step 7.2.2

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

$3 + \sqrt{4 (4) - 5} \leq 10$

Step 7.2.3

The left side $6.31662479$ is less than the right side $10$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $a > \frac{27}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 7.3.1

Choose a value on the interval $a > \frac{27}{2}$ and see if this value makes the original inequality true.

$a = 16$

Step 7.3.2

Replace $a$ with $16$ in the original inequality.

$3 + \sqrt{4 (16) - 5} \leq 10$

Step 7.3.3

The left side $10.68114574$ is greater than the right side $10$ , which means that the given statement is false.

False

Step 7.4

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

$a < \frac{5}{4}$ False

$\frac{5}{4} < a < \frac{27}{2}$ True

$a > \frac{27}{2}$ False

$a < \frac{5}{4}$ False

$\frac{5}{4} < a < \frac{27}{2}$ True

$a > \frac{27}{2}$ False

Step 8

The solution consists of all of the true intervals.

$\frac{5}{4} \leq a \leq \frac{27}{2}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$\frac{5}{4} \leq a \leq \frac{27}{2}$

Interval Notation:

$[\frac{5}{4}, \frac{27}{2}]$

Step 10