Problem 19 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(\frac{3 x-5}{2 x}\)

Step-by-Step Solution

Answer

For \(x=4\), the answer is \(\frac{7}{8}\). For \(x=6\), the answer is \(\frac{13}{12}\).

Step 1: Substitute the value of x (Part a)

Substitute the value of \(x = 4\) into the expression \(\frac{3x-5}{2x}\).

Step 2: Simplify the numerator (Part a)

Calculate \(3 \cdot 4 - 5\), which results in \(12 - 5 = 7\).

Step 3: Simplify the denominator (Part a)

Calculate the denominator \(2 \cdot 4 = 8\).

Step 4: Divide the results (Part a)

Divide the simplified numerator by the simplified denominator: \(\frac{7}{8}\).

Step 5: Substitute the value of x (Part b)

Substitute the value of \(x = 6\) into the expression \(\frac{3x-5}{2x}\).

Step 6: Simplify the numerator (Part b)

Calculate \(3 \cdot 6 - 5\), which results in \(18 - 5 = 13\).

Step 7: Simplify the denominator (Part b)

Calculate the denominator \(2 \cdot 6 = 12\).

Step 8: Divide the results (Part b)

Divide the simplified numerator by the simplified denominator: \(\frac{13}{12}\).

Key Concepts

substitution

Substitution is a fundamental concept in algebra where you replace a variable with a given value. This is extremely useful for evaluating expressions.
To substitute a value into an expression, simply replace each instance of the variable with the specified number.
For example, let's take the expression \(\frac{3x-5}{2x}\) and substitute \(x=4\):

• Replace \(x\) with 4: \(\frac{3(4)-5}{2(4)}\).

You can now proceed to simplify this new expression by performing the arithmetic operations.

simplifying expressions

Simplifying expressions involves performing basic arithmetic operations to combine like terms and make the expression as simple as possible.
After substitution, simplifying the expression \(\frac{3(4)-5}{2(4)}\) involves two main steps:

• First, simplify the numerator: Calculate \3 \times 4 - 5\. This results in \12 - 5 = 7\.
• Second, simplify the denominator: Calculate \2 \times 4\. This results in \8\.

Consequently, the simplified expression becomes \(\frac{7}{8}\).

division in algebra

Division in algebra is similar to dividing regular numbers but involves variables and their coefficients. Once you've simplified both the numerator and denominator, you perform the division operation.
Using our previous example, \(\frac{7}{8}\), simply divide the two numbers as you usually would:

• Here, you divide 7 by 8, which yields the fraction \(\frac{7}{8}\).

For division with different variables, always ensure that both the numerator and the denominator are fully simplified before performing the division.