Problem 37 - Algebra for College Students

Question

Solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. \(x+2<-3\) or \(x+2>3\)

Step-by-Step Solution

Answer

\((-\infty, -5)\) or \((1, \infty)\)

Step 1: Solve the first inequality

First, isolate the variable in the inequality \( x + 2 < -3 \). Subtract 2 from both sides to solve for \( x \):\[x + 2 - 2 < -3 - 2 \x < -5\]This gives us one of the solutions: \( x < -5 \).

Step 2: Solve the second inequality

Next, solve the other part of the compound inequality \( x + 2 > 3 \). Again, subtract 2 from both sides to get:\[x + 2 - 2 > 3 - 2 \x > 1\]This gives us the second solution: \( x > 1 \).

Step 3: Combine solutions for the compound inequality

The compound inequality is represented by the word 'or,' which means we combine the two solutions. The solution set is all \( x \) such that \( x < -5 \) or \( x > 1 \).

Step 4: Express in interval notation

To express the solution set \( x < -5 \) or \( x > 1 \) in interval notation:\[x < -5 \Rightarrow (-\infty, -5) \x > 1 \Rightarrow (1, \infty)\]Thus, the interval notation for the solution set is:\[(-\infty, -5) \, \text{or} \, (1, \infty)\]

Step 5: Graph the solution sets

On a number line, graph the solutions \( x < -5 \) and \( x > 1 \):1. \((-\infty, -5)\) is represented by a line starting from the left going to -5 with an open circle at -5.2. \((1, \infty)\) is represented by a line starting at 1 with an open circle and going to the right.Both intervals are graphed separately, indicating that no numbers between -5 and 1 are part of the solution.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is essential for visualizing solution sets on a number line. When graphing inequalities like \( x < -5 \) or \( x > 1 \), follow these simple steps:

• First, determine whether to use open or filled circles. For strict inequalities like \( < \) or \( > \), use open circles to indicate that the boundary point is not included.
• Next, draw a line on the number line to represent all the numbers that are part of the solution. For \( x < -5 \), the line starts at the open circle on -5 and extends indefinitely to the left.
• Similarly, for \( x > 1 \), place an open circle on 1 and draw the line extending to the right, showing that all numbers greater than 1 are included.

Visualizing inequalities helps you understand the range of numbers that are solutions.

Interval Notation

Interval notation is a compact way of describing a set of numbers along the real line. It is effective for expressing solutions to inequalities. Here's how it works:

• Use parentheses \( () \) for numbers that are not included in the solution set, relating to open circles on a graph.
• Use square brackets \( [] \) if a number is included, but in the case of our exercise, both solutions \( (-\infty, -5) \) or \( (1, \infty) \) use parentheses.
• The symbol \( -\infty \) or \( \infty \) is always accompanied by a parenthesis, because infinity is a concept, not a specific point you can reach or include.

For the compound inequality solutions such as \( x < -5 \) or \( x > 1 \), the interval notation captures all legitimate values in a succinct form.

Solving Inequalities

Solving inequalities involves isolating the variable of interest. When working with compound inequalities, tackle each part separately. Here's how you can solve a simple linear inequality:

• First, focus on isolating the variable. In \( x + 2 < -3 \), subtract 2 from both sides to get \( x < -5 \). Repeat this for each part of the compound inequality.
• Remember the special rule about multiplication or division by negative numbers in inequalities, though it doesn’t apply here: reverse the inequality sign.
• Pay attention to whether the inequality involves "or" or "and". "Or" means any solution that satisfies at least one part is acceptable, expanding the possible solutions.

Once solved, these inequalities give you a complete set of solutions expressed in terms of inequalities, graphs, and interval notation.