Problem 35 - Algebra for College Students

Question

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. \(x-2>-1 \quad\) and \(\quad x-2<1\)

Step-by-Step Solution

Answer

The solution is \((1, 3)\).

Step 1: Solve the first inequality

Let's solve the inequality \(x - 2 > -1\). Add 2 to both sides to isolate \(x\):\[x - 2 + 2 > -1 + 2\]Now, we have:\[x > 1\]

Step 2: Solve the second inequality

Now, solve the inequality \(x - 2 < 1\). Add 2 to both sides to isolate \(x\):\[x - 2 + 2 < 1 + 2\]Which simplifies to:\[x < 3\]

Step 3: Combine the inequalities

Since we have \(x > 1\) and \(x < 3\), we combine these to find the solution for the compound inequality. Combining these, we get:\[1 < x < 3\]

Step 4: Graph the solution set

To graph the solution set \(1 < x < 3\), draw a number line. Place open circles around 1 and 3 to indicate that these points are not included in the solution, and shade the line between these two points.

Step 5: Express the solution in interval notation

The interval notation for \(1 < x < 3\) is \((1, 3)\)This denotes all numbers greater than 1 and less than 3, but not including 1 and 3.

Key Concepts

Graphing Inequalities

Graphing inequalities involves illustrating the range of solutions on a number line. When we graph the inequality \(1 < x < 3\), we start by imagining a simple horizontal line that represents all possible values of \(x\). This line, called a number line, helps us identify which values solve the inequality.

For the inequality \(1 < x < 3\), we will:

• Locate the numbers 1 and 3 on the number line.
• Use open circles at 1 and 3 to show these endpoints are not included in the solution set. Open circles are used because the inequality does not allow equality, the solution only considers values strictly greater than 1 and less than 3.
• Shade the section of the number line between the two open circles to indicate all values greater than 1 and less than 3 are part of the solution.

This shading depicts visually which numbers satisfy the inequality, making it easier to understand how the ranges work in relation to the solved inequality.

Interval Notation

Interval notation is a shorthand way of writing the set of numbers that are solutions to an inequality. It simplifies communication by condensing an entire interval of numbers into a single expression. For our inequality \(1 < x < 3\), the solution set can be written in interval notation as \((1, 3)\).

Here’s how to interpret this notation:

• The parentheses \(()\) mean that the endpoints are not included in the interval. In our example, this tells us that 1 and 3 themselves are not solutions, which aligns with the open circles used in graphing.
• If the interval included the endpoints, square brackets \([]\) would be used. For example, if our inequality was \(1 \leq x \leq 3\), the interval notation would be \([1, 3]\), indicating that 1 and 3 are included.
• Interval notation is efficient for representing the range of x values that solve the problem.

This method is widely used because it's very concise and avoids the ambiguity that can arise with words or other notations.

Inequality Solving Steps

Solving compound inequalities involves a step-by-step approach to finding the range of values that satisfy both parts of the inequality. Let's break down the solution steps for the inequality \(x-2>-1\) and \(x-2<1\).

Step 1: Solve Each Inequality Separately

Start by isolating \(x\) in both inequalities:

• For \(x - 2 > -1\), add 2 to both sides to simplify to \(x > 1\).
• For \(x - 2 < 1\), similarly, add 2 to both sides to get \(x < 3\).

These results show individual constraints on \(x\), which need to be considered together.

Step 2: Combine the Solutions

To find a common solution that fits both inequalities, overlap the results from step 1. Combining \(x > 1\) and \(x < 3\) gives us the compound inequality \(1 < x < 3\).

Step 3: Verify Through Graphing and Notation

Always graph the solution on a number line and express it in interval notation. For this problem, we graph \(1 < x < 3\) with open circles and shade between them, and write the interval as \((1, 3)\).

Following these steps ensures the solution makes sense and fits the original problem accurately.