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Chapter 4: Problem 111

Write an equivalent inequality using absolute value. \(x\) is less than 2 units from 7 .

Short Answer

Expert verified
|x - 7| < 2

Step by step solution

01

Understand the Problem

The problem states that the distance between the variable x and 7 is less than 2 units. This can be translated into an absolute value inequality.
02

Define Absolute Value

The absolute value expression for the distance between x and 7 is \(|x - 7|\). This expression represents the distance between any number x and 7 on a number line.
03

Set Up the Inequality

Since the problem states that this distance is less than 2, we write the inequality as: \[|x - 7| < 2 \].

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Key Concepts

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

distance on a number line
Understanding the concept of distance on a number line is important for solving inequalities using absolute values. The distance between any two points on a number line is always a positive value or zero. For example, if we want to find the distance between -3 and 5, we calculate it as \(\text{distance} = |5 - (-3)| = |5 + 3| = 8\). This same principle is applied when dealing with absolute value inequalities where we measure the distance between a variable and a specific number.
absolute value
The absolute value of a number is its distance from zero on a number line, regardless of direction. It's always a non-negative value. So, \(|-4| = 4\) and \(|4| = 4\). When working with absolute value in inequalities, such as \(|x - 7| < 2\), we are looking at all the numbers x that are less than 2 units away from 7. Translating this idea visually helps in understanding why the inequality represents a certain range of values on the number line. This range is between 5 and 9, excluding the endpoints.
inequality solutions
Solving inequalities involves understanding the range of values that satisfy the inequality condition. For the inequality given in the exercise \(|x - 7| < 2\), it means we are looking for all x values that make the distance from 7 to x less than 2 units. To solve this, we break it into two parts: \(-2 < x - 7 < 2\). Solving these inequalities, we add 7 to each part, leading to \(5 < x < 9\). Thus, the solution to this inequality is that x can be any value between 5 and 9, not including 5 and 9 themselves. This approach ensures we find all values that meet the given condition on a number line.

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Most popular questions from this chapter

Graduate-School Admissions. Students entering a master's degree program at the University of Louisiana at Lafayette must meet minimum score requirements on the Graduate Records Examination (GRE). The GRE Verbal score must be at least 145 and the sum of the GRE Quantitative and Verbal scores must be at least \(287 .\) Each score has a maximum of \(170 .\) Using \(q\) for the quantitative score and \(v\) for the verbal score, write and graph a system of inequalities that represents all combinations that meet the requirements for entrance into the program.

For \(f(x)\) as given, use interval notation to write the domain of \(f\). $$ f(x)=\sqrt{8-2 x} $$

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ 12-|x-5| \leq 9 $$

Find the slope-intercept form of the equation of the line perpendicular to \(x-y=6\) that contains \((8,-9)\)

Solve and graph. Write the answer using both set-builder notation and interval notation. Let \(f(x)=5+|3 x-4| .\) Find all \(x\) for which \(f(x) \geq 16\)
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