Problem 24 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ -4 < 4 f+24 < 4 $$

Step-by-Step Solution

Step 1: Understanding the Inequality

The given inequality is a compound inequality: \(-4 < 4f+24 < 4\).This means we have two inequalities to work with:1) \(-4 < 4f+24\)2) \(4f+24 < 4\).

Step 2: Find the domain and intercepts

Determine the domain, x-intercepts, and y-intercepts.

Step 3: Analyze asymptotes and end behavior

Find vertical, horizontal, and oblique asymptotes.

Step 4: Find critical points and intervals

Compute the derivative, find critical points, and determine increase/decrease intervals.

Step 5: Summarize the graph

Combine all information to describe the graph.

Key Concepts

Compound Inequalities

Compound inequalities involve more than one inequality joined together. They are like two separate conditions that need to be satisfied at the same time. In this exercise, we have the compound inequality \(-4 < 4f+24 < 4\).

The goal is to manipulate each part to solve for the variable, which in this case is \(f\).
Compound inequalities come in two types: conjunctive and disjunctive.

• Conjunctive inequalities use the word "and", meaning both conditions must be true simultaneously. Our exercise is of this type, as shown by the double inequality symbol.
• Disjunctive inequalities use the word "or" and require at least one condition to be true. They represent a broader range of solutions.

By handling each smaller inequality separately and then combining the results, we can establish a full range of values that satisfy the compound inequality.

Graphing Solution Sets

After solving an inequality, the next step is to represent the solution graphically. This visualization helps in understanding which values satisfy the inequality.

For our exercise, we need to graph the solution set of the inequality \(-4 < 4f+24 < 4\) on a number line.
Here's how it works:

• First, solve each inequality part separately to find the potential solutions for \(f\).
• Once you have these solutions, they can be plotted on a number line.
• The solution set includes all values between the two boundaries determined in the solving step.

If an endpoint is included in the solution, it is marked with a solid dot. If the endpoint is not included, it is marked with an open circle. In the case of a strict inequality like our original one, where the value at the boundary is not included, open circles are used.

Number Lines

Number lines are a simple yet powerful tool to graphically represent solutions to inequalities. They consist of a horizontal line with evenly spaced markings or points that represent real numbers.

Using a number line to plot inequality solutions makes it easy to visualize which values are included in the solution set. Here’s how:

• Identify the integer points that are relevant to the inequality. Mark them clearly on the number line.
• Draw lines or segments to indicate the range of numbers that form the solution set.
• Use open circles for values not included and solid circles for included values.

Number lines help students quickly grasp which numbers satisfy an inequality, thus making them a favored method in algebra for representing solution sets.