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Chapter 17: Problem 31

Graph the given inequalities on the number line. $$x \leq-1 \text { or } x>0.5$$

Short Answer

Expert verified
Shade left of -1 with a solid dot at -1, and right of 0.5 with an open circle at 0.5.

Step by step solution

01

Understand the Inequalities

The first inequality is \( x \leq -1 \), which includes all numbers that are less than or equal to -1. The second inequality is \( x > 0.5 \), which includes all numbers greater than 0.5.
02

Plot the First Inequality

On a number line, plot \( x \leq -1 \) by drawing a solid dot at -1 to indicate that -1 is included, and shade all values to the left of -1. This represents all numbers less than or equal to -1.
03

Plot the Second Inequality

On the same number line, plot \( x > 0.5 \) by drawing an open circle at 0.5 (indicating that 0.5 is not included) and shade all values to the right of 0.5. This represents all numbers greater than 0.5.
04

Combine the Inequalities on the Number Line

Combine both plots on the same number line. The solution set for the inequality \( x \leq -1 \) or \( x > 0.5 \) will include all shaded areas from both plots. Thus, the solution covers two separate intervals: \((-\infty, -1]\) and \((0.5, +\infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Inequalities
When it comes to plotting inequalities on a number line, think of it as a way to visually represent all of the possible values that satisfy the given conditions. It's like painting a picture of what numbers will make the inequality true. To start, let's break down the concept:
  • An inequality symbol, like \( \leq \) (less than or equal to) or \( > \) (greater than), tells us which side of the number we're interested in for our solution.
  • A solid dot indicates that the number is part of the solution set. For instance, in \( x \leq -1 \), the \(-1\) is included in the solution set.
  • An open circle shows that the number is not part of the solution. For example, in \( x > 0.5 \), the \(0.5\) itself is not included, so we use an open circle at \(0.5\).
With these basics in mind, you can transform any inequality into a visual representation! It makes understanding which numbers solve the inequality much clearer.
Solution Sets
Solution sets are all about figuring out which numbers make the inequality true. It's like finding the answer to a puzzle, where each piece (number) fits into the solution we are trying to discover.
In the case of our exercise, we have two different inequalities to consider:
  • \( x \leq -1 \): This inequality includes all numbers less than or equal to \(-1\).
  • \( x > 0.5 \): This one includes all numbers greater than \(0.5\).
Now, when we say "or" in mathematics, it means we join all possible solutions from each inequality. Imagine collecting all the numbers that satisfy each inequality separately, then combining them. So, the solution set here has two separate parts: those satisfying \( x \leq -1 \) and those satisfying \( x > 0.5 \). Together, these sets form our complete solution: \((-\infty, -1]\) and \((0.5, +\infty)\).

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