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Chapter 4: Problem 106

Write an equivalent inequality using absolute value. $$ -5 \leq y \leq 5 $$

Short Answer

Expert verified
The equivalent inequality using absolute value is \( \left| y \right| \leq 5 \).

Step by step solution

01

Understand the original inequality

The given inequality is \( -5 \leq y \leq 5 \).This means that \( y \) is a number between -5 and 5 including -5 and 5.
02

Define the absolute value inequality

An absolute value inequality expresses the distance of a number from zero on the number line. For \( y \) to be between -5 and 5, it means the distance \( y \) from 0 should be no more than 5.
03

Write the absolute value inequality

To express that \( y \) is within 5 units of 0, we write: \[ \left| y \right| \leq 5 \]

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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 of a number from zero on the number line, regardless of direction. The absolute value of a number is always non-negative.
For example, the absolute value of both 5 and -5 is 5, expressed as \(|5| = 5\) and \(|-5| = 5\). This concept helps in understanding inequalities involving distance from zero.
In the given inequality \(-5 \leq y \leq 5\), the absolute value can be used to express this range as a single statement: \(|y| \leq 5\). This tells us that the distance from 0 to y is less than or equal to 5.
compound inequalities
Compound inequalities involve two or more inequalities combined into one statement with 'and' or 'or'.
The given exercise involves an 'and' compound inequality: \(-5 \leq y \leq 5\).
This means y must satisfy both inequalities at the same time, placing it between -5 and 5.
Using absolute value is another way to write this compound inequality. Hence, \(|y| \leq 5\) and \(-5 \leq y \leq 5\) convey the same range because both describe values of y within 5 units of 0.
number line
A number line helps to visualize inequalities and understand absolute value.
It's a horizontal line with numbers placed at equal intervals.
On a number line, 0 is the center, with positive numbers to the right and negative numbers to the left.
For the inequality \(-5 \leq y \leq 5\), marking -5 and 5 on the number line shows that y can be any number between these points, including -5 and 5.
By understanding this, it's easier to see that \(|y| \leq 5\) represents the distance from 0 to y being no more than 5 units, consistent with our number line visualization.

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Most popular questions from this chapter

Write an equivalent inequality using absolute value. $$ x<-4 \text { or } 4

Beverages. As sales of soft drinks decrease in the United States, sales of coffee are increasing. The revenue from sales of soft drinks, in billions of dollars, is approximated by $$ s(t)=0.33 t+17.1 $$ and the revenue from the sales of coffee, in billions of dollars, is approximated by $$ c(t)=0.6 t+9.3 $$ For both functions, \(t\) represents the number of years after \(2010 .\) Using an inequality, determine those years for which there will be more revenue from the sale of coffee than from soft drinks.

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

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The formula $$ C=\frac{5}{9}(F-32) $$ is used to convert Fahrenheit temperatures \(F\) to Celsius temperatures \(C\). a) Gold is liquid for Celsius temperatures \(C\) such that \(1063^{\circ} \leq C<2660^{\circ} .\) Find a comparable inequality for Fahrenheit temperatures. b) Silver is liquid for Celsius temperatures \(C\) such that \(960.8^{\circ} \leq C<2180^{\circ} .\) Find a comparable inequality for Fahrenheit temperatures.
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