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Chapter 8: Problem 72

Solve and graph. $$ |x-4|+5>2 $$

Short Answer

Expert verified
The solution is all real numbers. The entire number line is shaded.

Step by step solution

01

Isolate the Absolute Value Expression

First, subtract 5 from both sides of the inequality to isolate the absolute value expression: \[ |x-4| + 5 - 5 > 2 - 5 \]This simplifies to: \[ |x-4| > -3 \]
02

Analyze the Inequality

Since the expression inside the absolute value is always non-negative, any absolute value will always be greater than or equal to 0. Hence, the inequality \[ |x-4| > -3 \] will always be true for all real numbers because \(-3\) is a negative number.
03

Conclusion and Graphing

The inequality holds true for all real numbers. No matter what value of \(x\) is chosen, the expression \( |x-4| \) will always be greater than \(-3\). To graph this, draw a number line and shade the entire line, indicating that every real number is a solution.

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Key Concepts

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

absolute value
The absolute value of a number is its distance from 0 on the number line, regardless of direction. It is always non-negative. For example, \[ |3| = 3 \] and \[ |-3| = 3 \]. Absolute values are encased in vertical bars, like \(|x|\). When you see something like \(|x-4|\), it indicates the distance between \(x\) and 4.
In the exercise, the absolute value expression \(|x-4|\) means we're looking for how far \(x\) is from 4.
inequalities
Inequalities show the relationship between two expressions. They can tell us that one expression is less than, greater than, or fewer/greater than or equal to another. Common inequality symbols include:
  • \(<\) - less than
  • \(\leq\) - less than or equal to
  • \(>\) - greater than
  • \(\geq\) - greater than or equal to

In our exercise, we encountered \( |x-4| + 5 > 2 \). This is an absolute value inequality, and we want to find all values of \(x\) that satisfy this inequality.
graphing inequalities
Graphing inequalities helps to visualize the solution. For a simple inequality like \(x > 3\), you'd draw a number line, place an open circle at 3 (since 3 is not included), and shade everything to the right of 3.
In our exercise, after simplifying the inequality, we concluded that \( |x-4| > -3 \). Any absolute value is always non-negative, so this inequality is true for all real numbers. To represent this, we draw a number line and shade the entire line, indicating that every real number from \(-\infty\) to \(\infty\) is a solution.

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Most popular questions from this chapter

Classify each of the following statements as either true or false. The solution of \((x-1)(x-6)>0\) is \(\\{x | x<1 \text { or } x>6\\}\).

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Solve. $$ x^{2}+x-2<0 $$

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