Problem 57 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x<4 \text { or } x<-2 $$

Step-by-Step Solution

Answer

The solution set is \[ (-\infty, 4) \].

Step 1 - Understand the compound inequality

Identify that the compound inequality is joined by the word 'or', which means that the solution set includes values that satisfy either one of the inequalities.

Step 2 - Solve the first inequality

Solve the inequality \( x < 4 \) by leaving it as is since it is already in its simplest form. This condition includes all values of \( x \) that are less than 4.

Step 3 - Solve the second inequality

Solve the inequality \( x < -2 \) by leaving it as is since it is already in its simplest form. This condition includes all values of \( x \) that are less than -2.

Step 4 - Combine the solution sets

Since the inequalities are connected by 'or', combine the solution sets. The final solution set includes all values of \( x \) that are either less than 4 or less than -2. Any \( x \) value fulfilling one or both inequalities is part of the solution.

Step 5 - Express the solution in interval notation

Write the interval notation for the combined solution set. Since \( x < 4 \) includes values from -∞ to 4, and \( x < -2 \) includes values from -∞ to -2. The union of these intervals is \[ (-\infty, 4) \].

Step 6 - Graph the solution on a number line

Draw a number line and shade all regions to the left of 4, indicating all values less than 4. Make sure to include an open circle at 4, since 4 itself is not included in the solution.

Key Concepts

Solving Inequalities

To solve compound inequalities, we need to address each inequality separately. In the given example, we have two simple inequalities: \( x<4 \) and \( x<-2 \), connected by the word 'or.' This means that the solution set includes any value of \( x \) that satisfies either one of the inequalities.

Here’s how you do it step-by-step:

• First, look at **\( x<4 \)**. This inequality means that \( x \) can be any number less than 4. It doesn’t include 4 itself.
• Second, look at **\( x<-2 \)**. This tells us that \( x \) can be any number less than -2.

Now, because these two conditions are connected by 'or,' we need to combine the solution sets.

For 'or' statements, as long as \( x \) satisfies at least one of the inequalities, it is included in the solution set. In this example, every number less than 4 meets at least one of the inequalities \( x<4 \) or \( x<-2 \), so they all belong to the final solution set.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It’s a concise way to show which numbers are included in a set. Let's see how this applies to our compound inequality:

• **For \( x<4 \)**, we write this as \( (-fty, 4) \). This means all numbers less than 4.
• **For \( x<-2 \)**, we write this as \( (-fty, -2) \). This means all numbers less than -2.

For 'or' inequalities, we essentially unite these intervals, meaning we combine them into one interval that includes all solutions of both. In this case, the interval notation is also \( (-fty, 4) \), because any number less than 4 will satisfy at least one of the original inequalities.

Graphing Inequalities

Graphing inequalities visually represents the solutions on a number line, making it easier to understand. For the compound inequality \( x<4 \) or \( x<-2 \), start by drawing a number line.

• First, locate the number 4 on the number line and draw an open circle around it. The open circle indicates that 4 is **not** included in the solution.
• Next, shade all the numbers to the left of 4. This shading means all numbers less than 4 are part of the solution set.

Since the graph of \( x<-2 \) (another open circle at -2 and shading to the left) is already included in the graph of \( x<4 \), you don’t need to draw that part separately. All values less than 4, including those less than -2, are naturally included.
A complete graph helps you visually verify your solution and makes it easier to understand the range of values included in the solution set.