Problem 76 - A Graphical Approach to College Algebra

Question

In Exercises 75 and \(76,\) f and g are linear functions. If the solution set of \(f(x)

Step-by-Step Solution

Answer

(a) \(x = 3\); (b) \([3, \infty)\); (c) \((\infty, 3]\).

Step 1: Understanding the Inequality

The inequality \(f(x) < g(x)\) has a solution set of \((-fty, 3)\). This means that the function \(f(x)\) is less than \(g(x)\) for all \(x\) values less than 3.

Step 2: Solve for \(f(x) = g(x)\)

The equality \(f(x) = g(x)\) occurs at the boundary where \(f(x) < g(x)\) stops being true. According to the solution set \((\infty, 3)\), \(f(x) = g(x)\) at \(x = 3\).

Step 3: Solve for \(f(x) \geq g(x)\)

The inequality \(f(x) \geq g(x)\) is the opposite of \(f(x) < g(x)\). Since \(f(x) < g(x)\) is true for all \(x < 3\), \(f(x) \geq g(x)\) must be true for all \(x \geq 3\). Thus, the solution is \([3, \infty)\).

Step 4: Solve for \(f(x) \leq g(x)\)

This inequality is asking for when \(f(x)\) is less than or equal to \(g(x)\). We know \(f(x) < g(x)\) for \((\infty, 3)\) and \(f(x) = g(x)\) at \(x = 3\). Therefore, \(f(x) \leq g(x)\) for \((\infty, 3]\).

Key Concepts

Inequalities

Inequalities are statements that compare two values or expressions to denote one as larger or smaller than the other. They are essential in mathematics for expressing ranges of possible values rather than precise numbers. In the case of linear functions, inequalities often look at when one function surpasses another.
For instance:

• \(f(x) < g(x)\) shows that \(f(x)\) is smaller than \(g(x)\) within a range of \(x\) values.
• The direction of the inequality symbol (\(<, >, \leq, \geq \)) specifies the kind of relationship.

The inequality \(f(x) < g(x)\) implies that for all \(x\) values included in its solution set, \(f(x)\) lies below \(g(x)\) on a graph. Understanding inequalities aids in determining how functions compare across a spectrum of values.

Solution Sets

A solution set is the collection of all possible values that satisfy a given inequality or equation. These are often represented in interval notation, which provides a concise way to describe a range of solutions. For linear inequalities, the solution set showcases where on the number line the inequality holds true.

• An open interval, such as \((a, b)\), indicates that the values exactly at \(a\) and \(b\) are not part of the solution set.
• A closed interval, like \([a, b]\), includes both boundary points \(a\) and \(b\).

In the case of \(f(x) < g(x)\) with a solution set of \((-\infty, 3)\), it means that for every \(x\) less than 3, the inequality holds true. The understanding of how these intervals are constructed is key to solving and graphing inequalities.

Boundary Points

Boundary points mark the transition between different regions on the number line where various relationships between equations hold. These points are crucial in distinguishing when an inequality changes from true to false, or vice versa.
For instance, when determining where \(f(x) = g(x)\), you'd identify this at boundary points. When the inequality shifts, as in \(f(x) < g(x)\) to \(f(x) > g(x)\), that change occurs at the boundary point. At these transitions, we often find equality, such as \(x = 3\) here, which is a boundary point that turns \(f(x) < g(x)\) into \(f(x) = g(x)\).

• Identification of these points requires setting the expressions equal to find the exact value of \(x\).
• They help in determining whether a solution set is open or closed around specific values.

Therefore, understanding boundary points is instrumental for solving inequalities accurately.