Chapter 7.7, Problem 4BCYP

To determine

To Solve:

  $\sqrt{4x-4}-2<4$

Answer to Problem 4BCYP

Solution region is

  $2\le x<10$

Explanation of Solution

Given Information:

Given inequality:

  $\sqrt{4x-4}-2<4$

Concept Used :

Solve Radical inequalities:

• If the index of the root is even , identify the values of the variable for which the radicand is nonnegative.
• Solve the inequality algebraically.
• Test values to check.

Calculation:

• Given $\sqrt{4x-4}-2<4$

  $\begin{array}{l}\text{Since the radicand of a square root must be greater than or equal to zero, }\\ \text{first solve 4x-4}\ge 0\text{ to identify the values of }x\text{ for which the left hand side}\\ \text{of the inequality is defined}\text{.}\\ 4x-4\ge \mathrm{0....................}[\text{set the radicand }\ge \text{ 0]}\\ 2x\ge 0+\mathrm{4....................}[add\text{ 4 each side]}\\ \frac{2x}{2}\ge \frac{4}{2}\mathrm{........................}[\text{divide each side by 2]}\\ x\ge 2\end{array}$

  $\begin{array}{l}Now,solve\\ \sqrt{4x-4}-2<4\\ \sqrt{4x-4}<4+\mathrm{2.................}[\text{isolate the radical]}\\ \sqrt{4x-4}<6\\ {\left(\sqrt{4x-4}\right)}^2<6^2\mathrm{................}[\text{eliminate the radical]}\\ 4x-4<36\\ 4x<36+\mathrm{4.........................}[add\text{ 4 each side]}\\ 4x<40\\ x<\frac{40}{4}\mathrm{.............................}[\text{divide each side by 4]}\\ x<10\end{array}$

  $\begin{array}{l}Combining\\ x\ge 2\\ and\\ x<10\\ \text{The solution region is }2\le x<10,\text{ the intersection of the two regions}\text{.}\\ \text{We can check by taking a value from the region :}\\ x=5\\ \sqrt{4(5)-4}-2=\sqrt{20-4}-2=\sqrt{16}-2=4-2=2<4\\ So,\text{ the value from the region satisfies the inequality}\text{.}\\ \text{Hence, the solution region is }2\le x<10\end{array}$