Problem 8 - A Graphical Approach to College Algebra

Question

Find the zero of the function \(f\) $$f(x)=-8 x+0.5(2 x+8)$$

Step-by-Step Solution

Answer

The zero of the function is \(x = \frac{4}{7}\).

Step 1: Simplify the equation

First, we need to simplify the function expression: \[-8x + 0.5(2x + 8)\]Distribute the 0.5 inside the parentheses:\[0.5 imes 2x + 0.5 imes 8 = x + 4\]Substitute back into the equation:\[-8x + x + 4\]

Step 2: Combine like terms

Combine the terms involving \(x\):\[-8x + x = -7x\]Now the equation is simplified to:\[-7x + 4\]

Step 3: Set the function equal to zero

To find the zero of the function, set the equation equal to zero:\[-7x + 4 = 0\]

Step 4: Solve for x

Isolate \(x\) by subtracting 4 from both sides:\[-7x = -4\]Now, divide both sides by -7:\[x = \frac{-4}{-7} = \frac{4}{7}\]

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in mathematics. It involves reducing expressions to a simpler or more efficient form, making them easier to work with in calculations or when evaluating equations. In our exercise, the initial step is to simplify the function \[-8x + 0.5(2x + 8)\].
To do this, we employ the distributive property, a rule that allows us to remove parentheses by distributing a number across terms within the brackets.

• Multiply each term within the parentheses by 0.5: \[0.5 \times 2x + 0.5 \times 8 = x + 4\].
• We then rewrite the function, substituting the simplified expression: \[-8x + x + 4\].

This simplification helps us move smoothly into the process of finding zeros or roots of the function. Mastering this step is foundational, as it often streamlines problem-solving in various algebraic contexts.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. These equations are typically first-degree equations, meaning the highest power of the variable is one.
Consider the revised form of our equation after simplification: \[-7x + 4 = 0\].
To solve for the unknown variable \(x\), we follow a series of steps:

• First, we isolate the term containing \(x\) by subtracting 4 from both sides: \[-7x = -4\].
• Next, to solve for \(x\), divide both sides by -7, yielding \[x = \frac{4}{7}\].

Solving linear equations like this is crucial, as it forms the basis for solving more complex equations or discovering the solutions that make function expressions equal to zero. This operation is also essential for various applications in mathematics and the sciences.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions by merging terms that have common variables raised to the same power. This process reduces the number of terms and simplifies expressions.
In our function, the expression \[-8x + x + 4\] requires combining like terms:

• The terms \(-8x\) and \(x\) both involve the variable \(x\). These are combined by adding their coefficients: \(-8x + 1x = -7x\).

The outcome is a streamlined function expression: \[-7x + 4\].
Mastering the skill of combining like terms allows for more efficient and error-free simplification, and it is essential when solving equations and analyzing expressions. It’s a fundamental concept in algebra that provides clarity and accuracy in mathematical computations.