Proving an absolute value inequality

In summary, the conversation discusses the definition of absolute value, where it is equal to the value if the number is positive and negative of the value if the number is negative. Based on this definition, if $\left| a \right| \le b$, then $-b \le a \le b$. This can be proven using number lines and is divided into two cases: Case I where $a \ge 0$ and $\left| a \right| = a > b$, and Case II where $a < 0$ and $\left| a \right| = -a < b$. The difficulties mentioned are understanding the logical structure of the proof and a minor error in Case I.
  • #1
cbarker1
Gold Member
MHB
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If $\left| a \right| \le b$, then $-b\le a\le b$.
Let $a,b \in\Bbb{R}$ The definition of the absolute value is $ \left| x \right|= x, x\ge 0$ and $\left| x \right|=-x, x< 0$, where x is some real number.

Case I:$a\ge 0$, $\left| a \right|=a>b$

Case II: a<0, $\left| a \right|=-a<b$the solution is $-b<0\le a\le b$

I work on a number line. yet I still have trouble with the proof.
 
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  • #2
At which point of the proof are you facing difficulties?
 
  • #3
Well, I, for one, have trouble understanding the proof since I don't follow its logical structure, and not just because $a>b$ should be $a\le b$ in case I. Maybe someone can write a more coherent proof.
 

FAQ: Proving an absolute value inequality

What is an absolute value inequality?

An absolute value inequality is an inequality that contains an absolute value expression, which is denoted by two vertical lines surrounding a number or variable. It represents the distance of a number from 0 on a number line.

How do you solve an absolute value inequality?

To solve an absolute value inequality, you must isolate the absolute value expression on either side of the inequality symbol and then remove the absolute value bars. This will create two separate inequalities that you can solve for the variable.

How do you graph an absolute value inequality?

To graph an absolute value inequality, you must first rewrite the inequality in slope-intercept form. Then, you can plot the y-intercept and use the slope to plot additional points. Finally, draw a solid line for the graph and shade the appropriate region based on the inequality symbol.

What are the rules for solving absolute value inequalities?

The rules for solving absolute value inequalities include isolating the absolute value expression, removing the absolute value bars by creating two separate inequalities, and applying the appropriate inequality symbol based on the direction of the absolute value expression.

What are some real-world applications of absolute value inequalities?

Absolute value inequalities can be used to solve real-world problems involving distance, such as finding the range of possible values for a given measurement or determining the maximum or minimum distance between two points. They can also be used in physics to model situations involving velocity and acceleration.

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