Solve for a square root of 2a-3<5

$\sqrt{2 a - 3} < 5$

Step 1

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

${\sqrt{2 a - 3}}^{2} < 5^{2}$

Step 2

Simplify each side of the inequality.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 a - 3}$ as ${(2 a - 3)}^{\frac{1}{2}}$ .

${({(2 a - 3)}^{\frac{1}{2}})}^{2} < 5^{2}$

Simplify the left side.

Simplify ${({(2 a - 3)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 a - 3)}^{\frac{1}{2}})}^{2}$ .

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

${(2 a - 3)}^{\frac{1}{2} \cdot 2} < 5^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 a - 3)}^{\frac{1}{2} \cdot 2} < 5^{2}$

Rewrite the expression.

${(2 a - 3)}^{1} < 5^{2}$

Simplify.

$2 a - 3 < 5^{2}$

Simplify the right side.

Raise $5$ to the power of $2$ .

$2 a - 3 < 25$

Step 3

Solve for $a$ .

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

Add $3$ to both sides of the inequality.

$2 a < 25 + 3$

Add $25$ and $3$ .

$2 a < 28$

Divide each term in $2 a < 28$ by $2$ and simplify.

Divide each term in $2 a < 28$ by $2$ .

$\frac{2 a}{2} < \frac{28}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} < \frac{28}{2}$

Divide $a$ by $1$ .

$a < \frac{28}{2}$

Simplify the right side.

Divide $28$ by $2$ .

$a < 14$

Step 4

Find the domain of $\sqrt{2 a - 3} - 5$ .

Set the radicand in $\sqrt{2 a - 3}$ greater than or equal to $0$ to find where the expression is defined.

$2 a - 3 \geq 0$

Solve for $a$ .

Add $3$ to both sides of the inequality.

$2 a \geq 3$

Divide each term in $2 a \geq 3$ by $2$ and simplify.

Divide each term in $2 a \geq 3$ by $2$ .

$\frac{2 a}{2} \geq \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} \geq \frac{3}{2}$

Divide $a$ by $1$ .

$a \geq \frac{3}{2}$

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

$[\frac{3}{2}, \infty)$

Step 5

Use each root to create test intervals.

$a < \frac{3}{2}$

$\frac{3}{2} < a < 14$

$a > 14$

Step 6

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 $a < \frac{3}{2}$ to see if it makes the inequality true.

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

$a = 0$

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

$\sqrt{2 (0) - 3} < 5$

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

False

Test a value on the interval $\frac{3}{2} < a < 14$ to see if it makes the inequality true.

Choose a value on the interval $\frac{3}{2} < a < 14$ and see if this value makes the original inequality true.

$a = 4$

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

$\sqrt{2 (4) - 3} < 5$

The left side $2.23606797$ is less than the right side $5$ , which means that the given statement is always true.

True

Test a value on the interval $a > 14$ to see if it makes the inequality true.

Choose a value on the interval $a > 14$ and see if this value makes the original inequality true.

$a = 16$

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

$\sqrt{2 (16) - 3} < 5$

The left side $5.3851648$ is not less than the right side $5$ , which means that the given statement is false.

False

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

$a < \frac{3}{2}$ False

$\frac{3}{2} < a < 14$ True

$a > 14$ False

$a < \frac{3}{2}$ False

$\frac{3}{2} < a < 14$ True

$a > 14$ False

Step 7

The solution consists of all of the true intervals.

$\frac{3}{2} < a < 14$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$\frac{3}{2} < a < 14$

Interval Notation:

$(\frac{3}{2}, 14)$

Step 9

Solve for a square root of 2a-3<5

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