Problem 7 - A Graphical Approach to College Algebra

Question

Find the zero of the function \(f\) $$f(x)=3 x+6(x-4)$$

Step-by-Step Solution

Answer

The zero of the function is at \(x = \frac{8}{3}\).

Step 1: Simplify the Expression

The given function is \(f(x) = 3x + 6(x - 4)\). Start by expanding the term \(6(x - 4)\). Use the distributive property \(a(b + c) = ab + ac\) to simplify this part of the expression. So, \(6(x-4) = 6 \cdot x - 6 \cdot 4 = 6x - 24\). Now substitute back into the function: \[f(x) = 3x + 6x - 24\].

Step 2: Combine Like Terms

Combine the like terms in the expression \(f(x) = 3x + 6x - 24\). The like terms \(3x\) and \(6x\) can be added together: \[f(x) = (3x + 6x) - 24 = 9x - 24\].

Step 3: Set the Function to Zero

To find the zero of the function, set \(f(x)\) equal to zero and solve for \(x\): \[9x - 24 = 0\].

Step 4: Solve the Equation

Now solve the equation \(9x - 24 = 0\). Start by adding 24 to both sides to isolate terms with \(x\): \[9x = 24\].Next, divide both sides by 9 to solve for \(x\): \[x = \frac{24}{9} = \frac{8}{3}\].

Step 5: Verify the Solution

To verify, substitute \(x = \frac{8}{3}\) back into the simplified function \(f(x) = 9x - 24\) and check if it equals zero:\[f\left(\frac{8}{3}\right) = 9\left(\frac{8}{3}\right) - 24 = 24 - 24 = 0\].This confirms the solution is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to simplify expressions. It states that multiplying a single term by a sum or difference inside parentheses can be done by distributing the multiplication to each term individually. In mathematical terms, for any numbers or expressions \( a \), \( b \), and \( c \), the distributive property is expressed as follows:

• \( a(b + c) = ab + ac \)
• \( a(b - c) = ab - ac \)

This property helps in expanding expressions, making them easier to work with. For example, in the original exercise, we use the distributive property to expand \( 6(x - 4) \) into \( 6x - 24 \). Without this step, it would be difficult to manage complex equations or functions. Remember that this property applies regardless of the numbers or terms involved; it simplifies multiplying a term across a balanced expression in parentheses.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra and involves breaking down complex algebraic expressions into simpler, more manageable forms. This process generally includes combining like terms and using mathematical properties, such as the distributive property, to make calculations easier.
In the given exercise, after using the distributive property, we get the expression \( 3x + 6x - 24 \). Simplification involves:

• Combining like terms \( 3x \) and \( 6x \), which gives \( 9x \).
• Leaving the constants, such as \(-24\), untouched until further steps.

Simplifying makes calculations faster and reduces the possibility of mistakes in further steps, such as solving equations or evaluating functions. Practicing this not only facilitates performing algebraic operations but also nurtures a deeper understanding of the structural nature of algebraic expressions.

Solving Linear Equations

Solving linear equations entails finding the variable that makes the equation true. In a linear equation, each variable is raised to the first power and there are no products of variables. The coefficients and constants are straightforward, making them among the simplest algebraic inquiries. Let's break down the steps, especially in the context of the original exercise.
The problem requires us to find the zero of the function \( f(x) = 9x - 24 \) by setting it equal to zero. Here’s how we solve it:

• Set the function equal to zero: \( 9x - 24 = 0 \)
• Isolate the term with \( x \) by adding 24 to both sides: \( 9x = 24 \)
• Solve for \( x \) by dividing both sides by 9: \( x = \frac{24}{9} = \frac{8}{3} \)

These steps are systematic and show the logical progression from expanding and simplifying the function to actually solving for the variable. Verifying the solution by substituting it back into the original function ensures accuracy and builds confidence in the result. Understanding how to solve linear equations effectively opens up many possibilities for more complex problem-solving in algebra and beyond.