Question

In Exercises 37–44, solve the inequality. $$ 4 \sqrt{x-2}>20 $$

Short answer

The solution to the inequality \(4 \sqrt{x-2}>20\) is \(x > 27\).

Step-by-step solution

Simplification of the Inequality

First, isolate the square root. This can be done by dividing both sides of \(4 \sqrt{x-2}>20\) by 4 to give: \(\sqrt{x-2}>5\)

Remove the Square Root

The square root can be removed by squaring both sides. This gives: \(x - 2 > 25\)

Solving for x

We then isolate \(x\) by adding 2 to both sides. This results in: \(x > 27\)

Check Solution

To be sure of our solution, let's check the value. Place \(28\) (a number greater than 27) into the original equation. \(4 \sqrt {28-2} = 4 \sqrt {26}\), which is greater than 20. So our solution is correct.

Key concepts

Understanding Square Root Inequalities

When dealing with square root inequalities like \(4 \sqrt{x-2} > 20\), the goal is to isolate the square root on one side.
This involves manipulation of the inequality to make it easier to solve.

• Start by simplifying the inequality to remove coefficients like the 4 in this case.
• Divide both sides by the same number (4 in our example) to isolate the square root.
• Once isolated, the inequality becomes \(\sqrt{x-2} > 5\).

It’s important to understand that solving square root inequalities often involves squaring both sides at some point. This is a crucial step that helps remove the square root, making it simpler to find a solution. Always remember to check your solution by substituting a value back into the original inequality to confirm its accuracy.

Solving Inequalities

Solving inequalities involves similar steps to solving equations, but with some additional rules.

• Once any square roots are removed, and the inequality is simplified, as we see with \(x-2 > 25\), we focus on isolating the variable \(x\).
• Add or subtract terms to both sides of the inequality as required. In the example, we add 2 to both sides to isolate \(x\), giving \(x > 27\).
• Always consider reversing the inequality sign when multiplying or dividing by a negative number. While not applicable in our given exercise, it's a critical rule to remember.

These steps will lead to finding the solution set for the inequality, which tells us the range of values that satisfy the inequality condition. A crucial final step is to test or verify the solution to ensure it's correct.

Algebraic Expressions

Algebraic expressions form the backbone of equations and inequalities, consisting of variables, constants, and operations.

• Understand your terms: terms could be constants (like numbers) or coefficients multiplying variables (such as \(4x\)).
• In our problem, \(x-2\) is part of an algebraic expression under a square root.
• These expressions can often be simplified by following standard arithmetic operations: addition, subtraction, division, and multiplication.

Algebraic manipulation is essential for solving inequalities. By understanding how to systematically alter these expressions while maintaining the balance of the inequality, you can find solutions that satisfy the required condition. Always remember to backtrack through your steps to confirm correctness.