Problem 11 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 14 , find all the values of \(x\) for which the number is real. $$ \sqrt{8 x-5} $$

Step-by-Step Solution

Answer

The number \(\sqrt{8x-5}\) is real for \(x \geq \frac{5}{8}\).

Step 1: Understand the Condition for Real Numbers

For the expression \(\frac{a}{b}\sqrt[]{8x - 5}\) to be a real number, the value under the square root (the radicand) must be non-negative. This means the radicand must be greater than or equal to zero: \[\frac{a}{b}\displaystyle 8x - 5 \geq 0\]

Step 2: Set Up the Inequality

Write the inequality derived from Step 1: \[8x - 5 \geq 0\]

Step 3: Solve the Inequality

Isolate \(x\) on one side of the inequality. Add 5 to both sides: \[8x \geq 5\]. Then, divide both sides by 8: \[x \geq \frac{5}{8}\]

Step 4: Interpret the Result

The solution to the inequality \(8x - 5 \geq 0\) is \(x \geq \frac{5}{8}\). Thus, the expression \(\sqrt{8x - 5}\) is real for any \(x \) greater than or equal to \( \frac{5}{8} \).

Key Concepts

Radicand

The term 'radicand' refers to the number or expression inside a radical symbol. In the exercise given, the radicand is the expression inside the square root: \(8x - 5\). Understanding this term is crucial in solving problems involving square roots and other radicals.
For the square root of a number to be a real number, the radicand must be non-negative. This means we need to ensure that the expression inside the square root is greater than or equal to zero. Expressing this mathematically: \(8x - 5 \geq 0\).
Solving this inequality allows us to find the range of values for \(x\) that will keep the radicand non-negative and thus make the square root a real number. This concept applies not only to square roots but all even root functions, as they have the same requirement for a non-negative radicand.

Solving Inequalities

Solving inequalities involves finding the values of variables that satisfy the inequality. It's similar to solving equations, but with an additional focus on the inequality sign (\(\<\,\>\,\leq\,\geq\) ). Let's walk through the steps to solve the inequality from the exercise.
First, we started with the inequality: \(8x - 5 \geq 0\).
The goal is to isolate \(x\) on one side. Begin by adding 5 to both sides: \(8x \geq 5\).
Next, divide both sides by 8 to solve for \(x\): \[\frac{8x}{8} \geq \frac{5}{8} \Rightarrow x \geq \frac{5}{8}\]
So, \(x\) must be greater than or equal to \(5\/8\) for the square root to yield a real number. Always remember to reverse the inequality sign if you multiply or divide by a negative number while solving—that's a key rule!

Real Numbers

Real numbers include all the numbers on the number line. This constitutes rational and irrational numbers, encompassing positive numbers, negative numbers, zero, fractions, and decimals. When dealing with inequalities and square roots, understanding real numbers is essential.
In the context of the given exercise, \(x\) represents a real number that makes the expression \(\sqrt{8x -5}\) a real value. Since a real number solution is needed, the radicand (8x - 5) must be non-negative; therefore, our solution focuses on determining the values of \(x\) that ensure this condition.
After determining that \(x\geq 5/8\), any value of \(x\) within the range of real numbers that satisfies this inequality will make \(\sqrt{8x -5}\) a real number. When you solve inequalities, ensure your solutions fall within the domain of real numbers unless specified otherwise.