Problem 28 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{\frac{2 x}{3}+5} ; 6$$

Step-by-Step Solution

Answer

The result of evaluating the function for \(x = 6\) and rounding to the nearest tenth is \(y = 3.0\)

Step 1: Substituting the Value of x

Substitute \(x = 6\) into the function \(y= \sqrt{\frac{2x}{3}+5}\). This gives: \(y = \sqrt{\frac{2(6)}{3} + 5}\)

Step 2: Simplify the Fraction in the Square Root

Simplify the fraction to get: \(y = \sqrt{4 + 5} = \sqrt{9}\)

Step 3: Calculate the Square Root

Calculate the square root: \(y = 3\)

Step 4: Round to the Nearest Tenth

Since there are no decimal points in the answer, rounding 3 to the nearest tenth gives: \(y = 3.0\)

Key Concepts

Square Root

The square root is a mathematical concept where you find a number that, when multiplied by itself, results in the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9. It is often represented using the radical symbol \( \sqrt{} \).
In the context of function evaluation, finding the square root is necessary when the function itself contains a square root operation. To solve these, you simply determine what number, when squared, would equal the expression under the radical.
When working with square roots, it's crucial to ensure the number under the square root (known as the radicand) is non-negative, especially when dealing with real numbers. This is because the square root of a negative number isn't defined within the set of real numbers, and you would use complex numbers instead.
With real numbers, remember that both positive and negative roots can exist, but typically the principal (positive) square root is used in most mathematical contexts.

Simplifying Expressions

Simplifying expressions is all about making math easier by reducing algebraic expressions to their simplest form.
This involves a combination of operations such as addition, subtraction, multiplication, division, and especially in some cases, resolving fractions to their simplest form.
In the original problem, simplifying the expression involves first carrying out the operation in the denominator and then simplifying the fraction within the square root.Here’s how you do it:

• Perform any arithmetic operations inside the expression. For example, in this exercise, calculate \( \frac{2 \times 6}{3} \) which equals 4. This is done before adding the 5 to get a total of 9, leading us to the next step of finding its square root.
• Always focus on simplifying within any parentheses or radicals first to make further calculations easier.
• Apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to help guide the simplification process.

By simplifying, we make math problems easier to interpret and solve.

Rounding Numbers

Rounding numbers means adjusting the digits of a number to make it simpler or more suitable for a given context, usually to a specified degree of accuracy such as the nearest ten, hundred, or in this case, the nearest tenth.
When rounding to the nearest tenth, look at the number in the hundredths place. If it’s 5 or higher, you increase the tenths digit by 1. If it’s less than 5, you leave the tenths digit as it is. For the function evaluation in this exercise:

• Since you ended up with 3 after calculating the square root, you'll need to express this as a decimal for rounding purposes.
• A number like 3, when rounded to the nearest tenth, typically becomes 3.0. This is because there are no additional significant figures beyond the whole number.
• This helps in contexts requiring decimal representation, ensuring that number precision is consistent with the requirements of the problem.

Substitution Method

The substitution method is a technique mainly used in algebra to solve equations and functions. It involves replacing variables with given numbers in an expression or function. This allows you to directly evaluate the function for specific values and solve more complex expressions step-by-step.
In the problem provided, substitution is the first step to solving the equation. Given the value \( x = 6 \), we replace the variable \( x \) in the function \( y = \sqrt{\frac{2x}{3}+5} \).
Here’s how to apply this method effectively:

• Identify the variable that needs to be replaced, in this case, it’s \( x \).
• Substitute \( 6 \) into the expression wherever \( x \) appears, simplifying the equation to \( y = \sqrt{\frac{2(6)}{3}+5} \).
• Ensure all arithmetic operations to replace the variable are accurate to avoid mistakes in further calculations.

The substitution method is a handy way to work out numerical problems when dealing with algebraic expressions, ensuring that every variable is accounted for with definite numbers.