(Tricky) Absolute Value Inequalities

In summary, the conversation is discussing the proof that √x is uniformly continuous on [0, ∞]. The first step involves showing that |x - x0| < ε2 implies -ε2 + x0 < x < ε2 + x0. This is because ε is greater than 0, so x0 - ε2 is less than x0, and x could be either to the left or right of x0. The second step involves explaining the case involving the orange box, which is not specified in the first step. Thank you.
  • #1
vertciel
63
0
Hello everyone,

I'm posting here since I'm only having trouble with an intermediate step in proving that

[tex] \sqrt{x} \text{ is uniformly continuous on } [0, \infty] [/tex].

1zfjwxs.png


By definition, [tex] |x - x_0| < ε^2 \Longleftrightarrow -ε^2 < x - x_0 < ε^2 \Longleftrightarrow -ε^2 + x_0 < x < ε^2 + x_0 [/tex]

1. How does this imply the inequality in red?

[tex] \text{ Since } ε > 0 \text{ then } x_0 - ε^2 < x_0 [/tex]

However, I do not know more about x0 vs x.

2. Also, how does the above imply the case involving the orange; what "else" is there?

Thank you very much!
 
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  • #2
The inequality |x - x0| < ε2 doesn't specify whether x is to the right of x0 or to the left of it. That's the reason for the two inequalities.
 
  • #3
Thank you for your response, Mark44.

Could you please explain the red box?
 
  • #4
vertciel said:
Thank you for your response, Mark44.

Could you please explain the red box?
It looks like that's exactly what he did !
 

1. What is an absolute value inequality?

An absolute value inequality is an inequality that involves an absolute value expression. The absolute value of a number is its distance from zero on a number line. In an absolute value inequality, the variable is usually inside the absolute value symbol and the inequality symbol is used to compare the expression to a number or another expression.

2. How do you solve absolute value inequalities?

To solve an absolute value inequality, you can follow these steps:

  1. Isolate the absolute value expression on one side of the inequality.
  2. If the number or expression on the other side of the inequality is positive, then you can simply remove the absolute value symbols and solve the resulting inequality.
  3. If the number or expression on the other side of the inequality is negative, then you have to flip the inequality symbol and solve the resulting inequality.
  4. Check your solution by plugging it back into the original inequality.

3. Can you have more than one solution for an absolute value inequality?

Yes, it is possible to have more than one solution for an absolute value inequality. This is because the absolute value of a number can be either positive or negative, so the equation or inequality can have two different solutions depending on the value of the variable inside the absolute value symbol.

4. What is the difference between absolute value equations and absolute value inequalities?

The main difference between absolute value equations and absolute value inequalities is that equations have an equals sign (=) while inequalities have an inequality symbol (<, >, ≤, or ≥). In solving an absolute value equation, you are finding the value of the variable that makes the equation true. In solving an absolute value inequality, you are finding the values of the variable that make the inequality true.

5. How can absolute value inequalities be used in real life?

Absolute value inequalities can be used in real life to represent situations where there is a range of possible solutions. For example, if you are planning a trip and want to stay within a certain budget, you can use an absolute value inequality to represent the cost of the trip. The variable in the inequality would represent the total cost, and the inequality would represent the range of acceptable costs. This can also be applied to other real-life scenarios such as temperature ranges, inventory levels, and stock prices.

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