Problem 51 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ 4.5 x-2>2.5 \text { or } \frac{1}{2} x \leq 1 $$

Step-by-Step Solution

Answer

The solution set covers all real numbers: \(( -\infty, \infty )\).

Step 1: Solve the first inequality

We start with the first part of the compound inequality, \(4.5x - 2 > 2.5\). First, add 2 to both sides to isolate the term with \(x\). This gives:\[ 4.5x > 4.5 \]Next, divide both sides by 4.5 to solve for \(x\):\[ x > 1 \]

Step 2: Solve the second inequality

Now, solve the second part of the compound inequality, \(\frac{1}{2}x \leq 1\). Multiply both sides by 2 to eliminate the fraction:\[ x \leq 2 \]

Step 3: Write the compound solution in words

The compound inequality was given as an "or" condition, which means a solution satisfies either one inequality or the other (or both). Therefore:- \(x > 1\) or \(x \leq 2\)

Step 4: Convert the compound solution to interval notation

For \(x > 1\), the interval notation is \((1, \infty)\). For \(x \leq 2\), the interval notation is \((-\infty, 2]\). Since this is an "or" condition, the union of these intervals is the entire set of real numbers. This is because \(x\) is either greater than 1 or less than or equal to 2.

Step 5: Graph the solution set

To graph the solution, draw a number line. Shade the region to the left of 2 including the point 2 (filled circle at 2) and shade the region to the right of 1 (open circle at 1). The solution includes all numbers as it covers the whole number line.

Key Concepts

Interval Notation

Interval notation is a way of representing the set of solutions for inequalities. It uses parentheses and brackets to describe the range of numbers that satisfy an inequality, and it is a concise and clear way to understand the span of values involved.
When using interval notation:

• Parentheses, \((\), indicate that an endpoint is not included in the solution set. This is used for strict inequalities like \(x > 1\) or \(x < 2\).
• Brackets, \([\), indicate that an endpoint is included in the solution set. This applies to inequalities like \(x \geq 1\) or \(x \leq 2\).

In the exercise, "or" was the connecting term between inequalities. This means we form a union of solutions. If we have \(x > 1\), its interval representation is \((1, \infty)\), and if \(x \leq 2\), we represent it as \((-\infty, 2]\). The union of these two intervals covers the entire number line, representing the set of all real numbers in this case.

Solution Set

A solution set is the collection of all possible values that satisfy a given inequality or equation. When working with compound inequalities, solution sets can become a bit complex due to the combination of conditions.
In a compound inequality, we often encounter two types of connectors: "and" & "or".

• "And" implies that solutions must satisfy both conditions at the same time.
• "Or" implies that a solution can satisfy either of the conditions, or even both.

The original problem—\(4.5 x - 2 > 2.5 \) or \( \frac{1}{2} x \leq 1\)—is joined by "or." As a result, each part of the inequality has its solution, which are \(x > 1\) and \(x \leq 2\), making the solution set span all real numbers. This happens because any number greater than 1 satisfies the first condition, and any number less than or equal to 2 satisfies the second condition.

Graphing Inequalities

Graphing inequalities involves drawing on a number line to visually represent the solution set. Let's explore how to correctly graph inequalities:
When you graph inequalities like \(x > 1\), you will:

• Draw an open circle at \(x = 1\) to indicate that 1 is not included.
• Shade the line extending to the right towards infinity to show all values greater than 1.

For \(x \leq 2\), you will:

• Draw a closed circle at \(x = 2\) to show that 2 is included in the solution.
• Shade the line extending to the left towards negative infinity to include all values less than or equal to 2.

For this compound inequality connected by "or," you will find that when both inequalities are graphed together, shading will cover the entire number line. Thus, the graph visually confirms that every real number is within the solution set.