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Chapter 1: Problem 98

Solve. Consider only integer replacements. $$2<|x|<5$$

Short Answer

Expert verified
The integer solutions are x = -4, -3, 3, 4.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality is given as $$2 < |x| < 5$$This means that the absolute value of x must be greater than 2 and less than 5.
02

Split the Inequality

Recall that the absolute value of x, $$|x|$$means that x can be either positive or negative. Thus, we can split the inequality into:1. $$2 < x < 5$$2. $$-5 < x < -2$$
03

Identify Integer Solutions for Positive x

In the range $$2 < x < 5$$, the possible integer values for x are:- 3- 4
04

Identify Integer Solutions for Negative x

In the range $$-5 < x < -2$$, the possible integer values for x are:- -3- -4
05

Combine Solutions

Combine all the integer solutions from both ranges:- 3, 4, -3, -4Therefore, the integer solutions are: $$x = -4, -3, 3, 4$$

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Key Concepts

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

absolute value
Understanding absolute value is crucial for solving problems involving inequalities such as the one given in the exercise. The absolute value of a number represents its distance from zero on the number line, regardless of direction. For instance, both \(-3\) and \(3\) have an absolute value of 3. This concept helps us understand that \(|x|\) in the given inequality \(2 < |x| < 5\) implies that \(x\) could be either positive or negative. Absolute value strips the number of its sign, focusing only on its magnitude. Thus, when you see an absolute value inequality, always remember it covers both positive and negative cases.
integer solutions
When solving for integer solutions in inequalities, it鈥檚 essential to identify the whole numbers that fit within the specified ranges. In our exercise, the ranges are \(2 < x < 5\) and \(-5 < x < -2\). For \(2 < x < 5\), the possible integer values are 3 and 4. For \(-5 < x < -2\), the possible integer values are -3 and -4. An integer solution is simply a whole number that satisfies the inequality. When combining solutions from both ranges, we get the final set: 3, 4, -3, and -4. These values are valid because they respect the original constraints of the inequality.
inequalities
Inequalities involve expressions that show one value is greater than or less than another. In the inequality \(2 < |x| < 5\), it means that the absolute value of \(x\) is between 2 and 5, but not including 2 and 5. Solving absolute value inequalities often involves splitting the inequality into two scenarios. One where \(x\) is positive and another where \(x\) is negative, as shown in the exercise鈥檚 steps. For \(2 < |x| < 5\), this splits into two inequalities: \(2 < x < 5\) and \(-5 < x < -2\). These sub-inequalities are then treated separately to find the complete set of solutions.

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Most popular questions from this chapter

Write a true sentence using either \(<,>,\) or \(=\). $$|-5| \square|-2|$$

Given that \(0.3 \overline{3}=\frac{1}{3}\) and \(0.6 \overline{6}=\frac{2}{3},\) express each of the following as a ratio of two integers. $$7.7 \overline{7}$$

Write an equivalent expression without using grouping symbols. $$ -\left(8 x^{3}-6 x+5\right) $$

If \(n>0, m>0,\) and \(n \neq m,\) classify each of the following as either true or false. $$ n(-n-m)=-n^{2}+n m $$

Given that \(0.3 \overline{3}=\frac{1}{3}\) and \(0.6 \overline{6}=\frac{2}{3},\) express each of the following as a ratio of two integers. $$5.5 \overline{5}$$
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