Problem 18 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(\frac{x+2}{5}\)

Step-by-Step Solution

Answer

For \(x=4\), the result is \(1.2\). For \(x=6\), the result is \(1.6\).

Step 1: Understand the Expression

The expression given is \(\frac{x+2}{5} \). We are tasked with evaluating this expression for two values: \(x=4 \) and \(x=6\).

Step 2: Substitute and Simplify for \( a \) \(x=4\)

Substitute \(x=4\) into the expression: \( \frac{4+2}{5}\). Simplify the numerator: \(4 + 2 = 6\). Now we have: \( \frac{6}{5} = 1.2\).

Step 3: Substitute and Simplify for \( b \) \(x=6\)

Substitute \(x=6\) into the expression: \( \frac{6+2}{5}\). Simplify the numerator: \(6 + 2 = 8\). Now we have: \( \frac{8}{5} = 1.6\).

Key Concepts

Substitution in Algebra

Substitution in algebra means replacing variables with their given numerical values. In our example, we substitute the value of x into the expression \(\frac{x+2}{5} \). We have two cases to evaluate: when \(x=4\) and when \(x=6\).
First, let's do the substitution for \(x=4\). Place 4 in place of x: \(\frac{4+2}{5}\).
For our second scenario with \(x=6\), substitute 6 where x is: \(\frac{6+2}{5}\).
Substitution helps convert the algebraic expression into a numerical one that we can then simplify further.

Simplifying Fractions

Simplifying fractions is crucial when working with algebraic expressions. Our goal is to reduce the fraction to its simplest form for easier interpretation.
After substitution, the original expression \(\frac{x+2}{5}\) becomes \(\frac{6}{5}\) when \(x=4\) and \(\frac{8}{5}\) when \(x=6\).
Both fractions are already in their simplest forms since the numerator and the denominator have no common factors other than 1.
When simplified, fractions give us a clearer, more manageable answer which allows us to easily identify the value.

Numerical Evaluation

Numerical evaluation involves computing the final value of an expression once the variables have been substituted.
Let's evaluate our substituted expressions. For \(x = 4\), we get \(\frac{6}{5} = 1.2\). For \(x = 6\), we have \(\frac{8}{5} = 1.6\).
These numerical evaluations give us definite numbers that represent the value of the original algebraic expression for given values of x.
This final step in the process turns abstract expressions into concrete numbers, making understanding much easier.