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Chapter 1: Problem 75

Decide whether each statement is true or false. \(|13-8| \leq|7-4|\)

Short Answer

Expert verified
False

Step by step solution

01

Absolute Value Calculation

First, calculate the absolute values in the inequality \(|13-8| \leq |7-4|\). Determine the difference inside each absolute value: \(13-8 = 5\) and \(7-4 = 3\).
02

Evaluate Absolute Values

Next, find the absolute values of the results from Step 1. Since both differences are positive, their absolute values remain unchanged: \( |5| = 5\) and \( |3| = 3\).
03

Compare the Absolute Values

Now, compare the absolute values obtained: \( 5 \leq 3\). Since 5 is not less than or equal to 3, the statement is false.

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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 a number is from zero on a number line. It is always non-negative.
To find the absolute value, you simply take the positive version of a number. For example, the absolute value of -7 is 7, because -7 is 7 units away from zero.
Mathematically, absolute value is denoted using vertical bars, like this: \( |x| \). Here's how it works:
  • If \( x \geq 0 \), then \( |x| = x \).
  • If \( x < 0 \), then \( |x| = -x \).
Let's apply this to our example.
For the expression \(|13-8|\):
  • Calculate the difference: \(13 - 8 = 5\)
  • Since 5 is positive, its absolute value is \( |5| = 5 \).
Now for the expression \(|7-4|\):
  • Calculate the difference: \( 7 - 4 = 3 \)
  • Since 3 is positive, its absolute value is \( |3| = 3\).
Inequalities
Inequalities show the relationship between two values or expressions. They tell us if one value is less than, greater than, less than or equal to, or greater than or equal to another value.
They are represented by the following symbols:
  • \( < \) means 'less than'
  • \( \leq \) means 'less than or equal to'
  • \( > \) means 'greater than'
  • \( \geq \) means 'greater than or equal to'

For the given inequality \( |13-8| \leq |7-4| \), determining if it's true or false involves comparing the values we obtained from evaluating the absolute values:
  • The absolute value of \(13-8\) is 5, so \( |13-8| = 5\).
  • The absolute value of \(7-4\) is 3, so \( |7-4| = 3\).
Now, we compare these values using the less than or equal to symbol:
  • \( 5 \leq 3 \).
This statement is false because 5 is not less than or equal to 3.
Number Line Comparison
Visualizing values on a number line can make it easier to understand comparisons and inequalities. A number line is a straight line on which every point represents a number.
In this case:
  • First, plot the number 5 and 3 on the number line. 5 will be to the right of 3, as it's larger.
  • Now, check the inequality \( 5 \leq 3 \) using their positions. Since 5 is to the right of 3, it means 5 is greater than 3.
When you see that 5 is to the right of 3, it becomes visually clear that 5 is not less than or equal to 3.
This visual understanding helps confirm that the statement \(|13-8| \leq |7-4|\), which simplifies to \(5 \leq 3 \), is false.

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Most popular questions from this chapter

Perform each indicated operation. $$ -9+[(3-2)-(-4+2)] $$

The two tables show the heights of some selected mountains and the depths of some selected trenches. Use the information given to answer. \( \)\begin{array}{|l|c|}\hline \text { Mountain } & {\text { Height (in feet) }} \\\ {\text { Foraker }} & {17,400} \\ {\text { Wilson }} & {14,246} \\\ {\text { Pikes Peak }} & {14,110} \\ \hline\end{array} $$ \begin{array}{|l|c|}\hline\text {} & {\text { Depth }( \text { in feet } } \\ {\text { Trench }} & {\text { as a negative number }} \\ {\text { Philippine }} & {-32,995} \\ {\text { Cayman }} & {-24,721} \\ {\text { Java }} & {-23,376} \\ \hline \end{array} $$ How much deeper is the Cayman Trench than the Java Trench?

Find each difference. $$ \frac{1}{3}-\left(-\frac{4}{3}\right) $$

The two tables show the heights of some selected mountains and the depths of some selected trenches. Use the information given to answer. \( \)\begin{array}{|l|c|}\hline \text { Mountain } & {\text { Height (in feet) }} \\\ {\text { Foraker }} & {17,400} \\ {\text { Wilson }} & {14,246} \\\ {\text { Pikes Peak }} & {14,110} \\ \hline\end{array} $$ \begin{array}{|l|c|}\hline\text {} & {\text { Depth }( \text { in feet } } \\ {\text { Trench }} & {\text { as a negative number }} \\ {\text { Philippine }} & {-32,995} \\ {\text { Cayman }} & {-24,721} \\ {\text { Java }} & {-23,376} \\ \hline \end{array} $$ What is the difference between the height of Pikes Peak and the depth of the Java Trench?

Write a numerical expression for each phrase and simplify. The sum of \(-5\) and 12 and 6.
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