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Chapter 1: Problem 29

Determine whether the given value of \(x\) satisfies the inequality. $$|x-2|>4 ; x=3$$

Short Answer

Expert verified
No, the given value of \(x\) does not satisfy the inequality.

Step by step solution

01

Compute the expression inside the absolute value

First, we must substitute the provided value of \(x\) into \(x-2\). This implies that if \(x=3\), we have \(3-2 = 1\). Hence, the expression inside the absolute value becomes \(1\).
02

Determine the absolute value

Now we substitute \(1\) into the absolute value. The absolute value of \(1\) is \(1\) (simply because the distance between \(1\) and \(0\) on the number line is \(1\)). So, the inequality becomes \(1 > 4\).
03

Evaluate the inequality

Evaluating the inequality \(1 > 4\) results in a False statement, because \(1\) is not greater than \(4\).

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Key Concepts

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

Absolute Value
The concept of absolute value is all about understanding distance on the number line. Absolute value measures how far a number is from zero, regardless of direction, so it is always positive or zero. For example, the absolute value of both -3 and 3 is 3, written as \(|-3| = 3\) and \(|3| = 3\), because both are three units away from zero.
When working with absolute values in algebra, remember these key points:
  • The absolute value of zero is zero: \(|0| = 0\).
  • For any positive number \(a\), its absolute value is \(a\): \(|a| = a\).
  • For any negative number \(b\), its absolute value is the positive of \(b\): \(|b| = -b\).
The absolute value helps express distance without considering direction, which is useful when solving inequalities involving absolute values.
Inequality Evaluation
Inequalities are mathematical expressions that show a relationship where one value is not equal to another. In this exercise, we have an inequality involving absolute value: \(|x - 2| > 4\).
Evaluating inequalities requires replacing the variables with numbers and determining if the statement holds true. For example, if \(x = 3\), we substitute it back into the inequality, leading to evaluating \(|3 - 2| > 4\). This simplifies to \(1 > 4\), which is false.
Key steps when evaluating inequalities:
  • Substitute the given value of the variable into the inequality.
  • Calculate any operations and simplify the expression.
  • Determine if the resulting statement accurately represents the inequality.
In our example, the inequality was evaluated as false, highlighting that \(x = 3\) does not satisfy it.
Mathematical Reasoning
Mathematical reasoning involves logically dissecting problems and step-by-step methods to solve them. In the context of this exercise, it includes analyzing the expression within the absolute value, calculating its absolute value, and evaluating the inequality.
Reasoning through inequalities with absolute values means understanding:
  • The nature of absolute values and how they transform the equation, turning negative values into positive.
  • That the inequality \(|x - 2| > 4\) means we're looking for numbers on the number line that are more than 4 units away from 2.
  • Interpreting results based on the calculations made.
This logical process demonstrates how one can methodically approach and solve absolute value inequalities, providing clarity on whether given values satisfy the inequality conditions.

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Most popular questions from this chapter

Applications In this set of exercises you will use the concepts of intersection of lines and linear inequalities to study real-world problems. Cost and Revenue In Exercises \(69-72,\) for each set of cost and revenue functions, (a) find the break-even point and (b) calculate the values of \(q\) for cohich retucnue exceeds cost. $$\begin{aligned} &C(q)=10 q+200\\\ &R(q)=15 q \end{aligned}$$

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