Problem 62 - Elementary Algebra

Question

For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number. $$ \sqrt{7 x+8} $$

Step-by-Step Solution

Answer

Answer: The restrictions on x for the expression \(\sqrt{7x+8}\) to be a real number are \(x \geq \frac{-8}{7}\).

Step 1: Identify the expression under the square root

We have the expression \(\sqrt{7x+8}\) and we need to find restrictions for x. First, we need to identify the expression under the square root, which is \(7x+8\).

Step 2: Set the expression greater than or equal to zero

To ensure that the square root remains a real number, the expression under the square root must be greater than or equal to zero. So we can write the inequality as follows: \(7x+8\geq0\)

Step 3: Solve the inequality for x

Now, we need to solve the inequality to find the restrictions on x. Subtract 8 from both sides: \(7x\geq-8\) Next, divide both sides by 7: \(x\geq\frac{-8}{7}\)

Step 4: Write the restrictions on x

We found that the value of x must be greater than or equal to \(\frac{-8}{7}\). So, the proper restrictions on x are: \(x \geq \frac{-8}{7}\)

Key Concepts

Understanding Square Roots

When we deal with square roots, we're looking at expressions that involve taking the square root of a number or variable. The square root of a number is another number, which when multiplied by itself gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In mathematical terms, it's written as \(\sqrt{9} = 3\).

In expressions involving variables, like \(\sqrt{7x + 8}\), you need to ensure that whatever is under the square root is positive or zero, to avoid imaginary numbers. That's because square roots of negative numbers aren't real. We'll explore why this is important in the next sections.

Exploring Inequalities

Inequalities help us understand the range of values a variable can take. An inequality like \(7x + 8 \geq 0\) means that the expression \(7x + 8\) should be greater than or equal to zero. This ensures that the square root results in a real number. You solve these inequalities just like equations, making sure to maintain the inequality sign throughout.

For example:

• Start with \(7x + 8 \geq 0\).
• Subtract 8 from both sides to get \(7x \geq -8\).
• Divide everything by 7 to isolate \(x\), resulting in \(x \geq \frac{-8}{7}\).

These steps guide us in finding the restrictions on the variable for the expression to remain valid in the set of real numbers.

Understanding Variable Restrictions

Variable restrictions refer to the specific conditions that a variable must satisfy. These restrictions are crucial when dealing with square roots. For the expression \(\sqrt{7x + 8}\) to yield a real number, the value under the square root must be zero or positive.

To identify the restriction, we set up an inequality:

• The expression \(7x + 8\) should be non-negative.
• Solve \(7x + 8 \geq 0\) to find \(x \geq \frac{-8}{7}\).

This means \(x\) needs to be larger than or equal to \(\frac{-8}{7}\). These restrictions ensure that the mathematical expression doesn't venture into regions where it would stop representing a real number.

Securing Real Number Solutions

Real number solutions are values that can be plotted on the number line. These are numbers without imaginary components. When dealing with square roots, ensuring the expression results in real numbers is key. By solving \(7x + 8 \geq 0\), we maintained real values under the square root.

This process gives us the domain:

• \(x \geq \frac{-8}{7}\)

This inequality tells us that any number equal to or larger than \(\frac{-8}{7}\) will work. It ensures all computed square roots stay within the realm of real numbers, providing valid, usable results within mathematical calculations.