Problem 77 - Intermediate Algebra

Question

Let \(f(x)=\frac{1}{2} x-\frac{2}{3}\) and \(g(x)=x+\frac{4}{3} .\) Find all values of \(x\) for which \(f(x)

Step-by-Step Solution

Answer

The solution set is \(x > -4\) or \((-4, \infty)\) in interval notation.

Step 1: Write the Inequality

To find the values of \(x\) for which \(f(x) < g(x)\), you first set up the inequality \(\frac{1}{2}x - \frac{2}{3} < x + \frac{4}{3}\).

Step 2: Isolate x on One Side

Subtract \(x\) from both sides of the inequality to get \(\frac{1}{2}x - x - \frac{2}{3} < \frac{4}{3}\). This simplifies to \(-\frac{1}{2}x - \frac{2}{3} < \frac{4}{3}\).

Step 3: Eliminate the Fraction Coefficient

Add \(\frac{2}{3}\) to both sides to isolate the term with \(x\), resulting in \(-\frac{1}{2}x < \frac{4}{3} + \frac{2}{3}\). Combine the fractions on the right side to get \(\frac{6}{3} = 2\). Now the inequality is \(-\frac{1}{2}x < 2\).

Step 4: Solve for x

Multiply both sides by \(-2\) to solve for \(x\), remembering to flip the inequality sign when multiplying by a negative number: \(x > -4\).

Step 5: Write the Solution Set

The solution set for \(x\) is all values greater than \(-4\). In interval notation, this is \((-4, \infty)\).

Key Concepts

Understanding Functions in Inequalities

Functions are fundamental elements in mathematics, helping us map inputs to outputs using a defined rule. In this exercise, two functions are provided: \( f(x) = \frac{1}{2}x - \frac{2}{3} \) and \( g(x)=x+\frac{4}{3} \). When you are tasked with finding values for which \( f(x) < g(x) \), it means you are comparing the outputs of these functions for different input values of \( x \).

• **\( f(x) \) signifies the position along a line based on the input \( x \); it is a linear function because it forms a straight line when graphed.**
• **Similarly, \( g(x) \) is another linear function with its own unique slope and y-intercept.**
• **The inequality \( f(x) < g(x) \) implies that for certain inputs, the output of \( f \) is less than the output of \( g \).**

Understanding how these functions interact, particularly when placed in an inequality, allows you to solve for values of \( x \) where this condition is true.

A Step-by-Step Guide to Solving Inequalities

To solve the inequality given in the exercise, we follow a series of logical steps, each bringing us closer to finding the acceptable range for \( x \).

• **Start by setting up the inequality:** Align \( \frac{1}{2}x - \frac{2}{3} < x + \frac{4}{3} \). This step ensures that you have a clear statement of the inequality.
• **Isolate the variable on one side:** This usually involves subtracting terms, as seen when we subtract \( x \) from both sides to simplify the expression: \( -\frac{1}{2}x - \frac{2}{3} < \frac{4}{3} \). The goal here is to gather all terms involving \( x \) on one side.
• **Combine and simplify:** Next, add \( \frac{2}{3} \) to both sides, allowing us to further simplify: \( -\frac{1}{2}x < 2 \).
• **Solve for \( x \):** The crucial final step is multiplying by \(-2\) to solve for \( x \), taking care to flip the inequality sign. This results in \( x > -4 \).

By following through these steps carefully, the result is a clearer understanding of the conditions under which one function is less than another.

Reading and Using Interval Notation

Interval notation provides a concise way to express a set of solutions or values that satisfy a certain condition. Upon solving the inequality, you arrived at \( x > -4 \). In interval notation, this is written as \((-4, \infty)\).

• **The parenthesis (**\((-4, \infty)\)**) indicate that \(-4\) is not included in the solution set, as \( x \) must be greater than, but not equal to, \(-4\).**
• **The infinity symbol (\(\infty\)) signifies that there is no upper bound on the value of \( x \).**

This notation is especially helpful in mathematics as it provides clarity and simplicity, making it easy to convey the range of possible values succinctly in a standard form. Able to quickly communicate solution sets, it becomes a powerful tool in both simple and complex equations.