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Chapter 2: Problem 43

Solve using the addition principle. Graph and write both set-builder notation and interval notation for each answer. $$ x-8 \leq-10 $$

Short Answer

Expert verified
\(x \leq -2\) in set-builder notation is \( \{x | x \leq -2\} \), and in interval notation, it is \( (-\infty, -2] \).

Step by step solution

01

- Identify the inequality

The given inequality is \(x - 8 \leq -10\). Our goal is to solve for x.
02

- Add 8 to both sides

To isolate x, add 8 to both sides of the inequality: \(x - 8 + 8 \leq -10 + 8\)
03

- Simplify the inequality

Simplify the expression: \(x \leq -2\)
04

- Write in set-builder notation

The solution in set-builder notation is: \( \{x | x \leq -2\} \)
05

- Write in interval notation

The solution in interval notation is: \( (-\infty, -2] \)
06

- Graph the solution

To graph the solution, draw a number line and shade all the values to the left of -2, including -2 itself. Use a closed dot on -2 to indicate that -2 is part of the solution.

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Key Concepts

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

addition principle
The addition principle is a fundamental concept in solving inequalities. It states that you can add the same number to both sides of an inequality without changing the inequality's direction. In our example, we start with the inequality: $$x - 8 \leq -10$$. To solve for \(x\), we need to isolate \(x\) on one side of the inequality. Here's how we do it:
1. Add 8 to both sides of the inequality to cancel out the -8 on the left side: $$x - 8 + 8 \leq -10 + 8$$.
2. Simplify: $$x \leq -2$$. The addition principle helps us isolate the variable \(x\) by eliminating the constant term, thus providing us with the solution \(x \leq -2\).
set-builder notation
Set-builder notation is a concise way of expressing the solution set of an inequality. It specifies all the possible values that satisfy the given inequality. In our example, after solving the inequality, we found: $$x \leq -2$$. To express this solution in set-builder notation, we write: \(\{x | x \leq -2\}\).
The vertical bar \(|\) is read as 'such that'. So this notation reads as 'the set of all \(x\) such that \(x\) is less than or equal to -2'. This notation is very useful in mathematical analysis because it clearly lays out the criteria for membership in the set.
interval notation
Interval notation provides a way to describe the set of solutions to an inequality using intervals. The interval notation for our solution \(x \leq -2\) is \((-\infty, -2]\).
This notation uses parentheses \(()\) and brackets \([]\) to indicate whether endpoints are included in the interval. Here:
  • \(-\infty\) represents negative infinity, indicating that the values extend infinitely in the negative direction.
  • Since -2 is included in the solution, we use a closing bracket \([]\).

So, \((-\infty, -2]\) tells us that the solution includes all numbers from negative infinity up to and including -2.
graphing inequalities
Graphing inequalities involves representing the solution set on a number line. For the inequality \(x \leq -2\), we:
  • Draw a number line.
  • Locate -2 on the number line.
  • Use a closed dot at -2 to indicate that -2 is part of the solution.
  • Shade the region to the left of -2 to represent all the values less than -2.
This visual representation helps us easily understand the range of values that satisfy the inequality. It shows at a glance that all numbers less than or equal to -2 are included in the solution set.

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