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Chapter 2: Problem 26

For Problems \(17-30\), solve each inequality and graph the solution. $$ |x+1| \leq 1 $$

Short Answer

Expert verified
The solution is the interval \([-2, 0]\) and is graphed as a shaded line segment between \(-2\) and \(0\) on the number line, including both endpoints.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality \(|x+1| \leq 1\) means that the distance between \(x+1\) and 0 is less than or equal to 1. This implies that \(-1 \leq x+1 \leq 1\).
02

Break Down the Compound Inequality

We need to solve the compound inequality \(-1 \leq x+1 \leq 1\). This involves two separate inequalities:1. \(-1 \leq x+1\)2. \(x+1 \leq 1\).
03

Solve the First Inequality

To solve \(-1 \leq x+1\), subtract 1 from both sides to obtain: \(x \geq -2\).
04

Solve the Second Inequality

To solve \(x+1 \leq 1\), subtract 1 from both sides to obtain: \(x \leq 0\).
05

Combine the Solutions

The solution to the compound inequality \(-1 \leq x+1 \leq 1\) is the intersection of the two sets of solutions. Therefore, \(-2 \leq x \leq 0\).
06

Graph the Solution

Draw a number line. Shade the region between \(x = -2\) and \(x = 0\), including the endpoints, as it is an inclusive inequality.

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Key Concepts

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

Absolute Value
Absolute value represents the distance of a number from zero on the number line regardless of direction. It is always a non-negative number. For the expression \(|x+1|\), the absolute value tells us how far \(x+1\) is from zero, making it a crucial part of solving the inequality \(|x+1| \leq 1\).

This specific inequality implies that the expression \(x+1\) is at most 1 unit away from zero. Remember that within an absolute value inequality, you often need to consider two scenarios for solving: the positive scenario and the negative scenario. Since distance is being measured, the inequality naturally splits into two parts to capture both directions on a number line.
Compound Inequality
A compound inequality involves connecting two inequalities with either "and" or "or". In this case, we worked with an "and" compound inequality: \(-1 \leq x+1 \leq 1\).

This expression means that both individual inequalities must be satisfied at the same time. So, we break it into two parts: \(-1 \leq x+1\) and \(x+1 \leq 1\). Each part is solved separately to find the range of values which satisfy both conditions concurrently.

This process is essentially examining the overlaps on the number line, ensuring that \(-2 \leq x \leq 0\) is a solution that works for both.
Graphing Inequalities
Graphing inequalities involves representing the solution on a number line to visually outline the set of values that satisfy the inequality. In the exercise, once we solved the inequality \(-2 \leq x \leq 0\), the solution was graphed on a number line.

Here are the steps to graphing:
  • Draw a number line with regular divisions.
  • Identify the boundary numbers \(x = -2\) and \(x = 0\).
  • Shade the region between these numbers to show all possible values of \(x\) that satisfy the inequality.
  • Since the inequality is "less than or equal to" or "greater than or equal to," we include the endpoints. Indicate this by drawing solid dots at \(x = -2\) and \(x = 0\).
Graphing serves as a visual confirmation that the solution set is accurate and easy to comprehend.

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Most popular questions from this chapter

Solve each problem by setting up and solving an appropriate inequality. Mona invests \(\$ 1000\) at \(8 \%\) yearly interest. How much does she have to invest at \(6 \%\) so that the total yearly interest from the two investments exceeds \$170?

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -7<3 x-1<8 $$

A person's intelligence quotient (I) is found by dividing mental age \((M)\), as indicated by standard tests, by chronological age \((C)\) and then multiplying this ratio by 100 . The formula \(\mathrm{I}=\frac{100 M}{C}\) can be used. If the I range of a group of 11-year-olds is given by \(80 \leq \mathrm{I} \leq 140\), find the range of the mental age of this group.

Find the solution set for each of the following compound statements, and in each case explain your reasoning. (a) \(x<3\) and \(5>2\) (b) \(x<3\) or \(5>2\) (c) \(x<3\) and \(6<4\) (d) \(x<3\) or \(6<4\)

Solve each problem by setting up and solving an appropriate inequality. Thanh has scores of \(52,84,65\), and 74 on his first four math exams. What score must he make on the fifth exam to have an average of 70 or better for the five exams?
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