Solve Inequality: x^2-4|-3|<1 | UofT Calculus

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In summary, when solving the inequality ||(x^2)-4|-3|< 1, you can either plot a graph of the function or split x into two cases: when x^2 ≤ 4 and when x^2 > 4. For the first case, you want |x^2-1| < 1 and for the second case, you need |x^2-7| < 1. These conditions will help you determine the possible values for x. It is important to show your attempt at solving the equation when asking for help.
  • #1
Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality:
||(x^2)-4|-3|< 1
 
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AnthonyJohnAre said:
Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality:
||(x^2)-4|-3|< 1

The easiest way is to first gain some insight into the problem by plotting a graph of the function f(x) =||x^2 - 4|-3|.

A systematic way is to split up x into cases: (1) x^2 ≤ 4; (2) x^2 > 4. In case (1) we have f(x) = |4-x^2-3| = |1-x^2| = |x^2-1|. So, for x^2 ≤ 4 we also want |x^2-1| < 1. What do these requirements say about x? In case (2) we have f(x) = |x^2 - 4 - 3| = |x^2-7|, so for x^2 > 4 we also need |x^2-7| < 1. What do these tell you about x?
 
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  • #3
AnthonyJohnAre said:
Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality:
||(x^2)-4|-3|< 1

Please check your PMs. You should show your attempt at solving the equation as part of your first post asking for help.
 

1. What is an inequality?

An inequality is a mathematical expression that compares two quantities using symbols such as <, >, ≤, ≥ to indicate which quantity is greater or less than the other.

2. How do you solve an inequality?

To solve an inequality, you need to isolate the variable on one side of the inequality symbol and simplify the other side. You may need to use inverse operations, combine like terms, and follow the rules of order of operations.

3. What does the absolute value symbol mean?

The absolute value symbol, denoted by |x|, represents the distance of a number from 0 on a number line. It always results in a positive value, regardless of the sign of the number inside.

4. What is the UofT Calculus notation for inequalities?

The UofT Calculus notation for inequalities is using the interval notation, which uses square brackets [ ] or parentheses ( ) to represent the inclusion or exclusion of the endpoint values. For example, [2, 5) means the interval from 2 to 5, including 2 but not including 5.

5. How do you solve the inequality x^2-4|-3|<1 | UofT Calculus?

To solve this inequality, we first need to rewrite it using the absolute value notation. Since |-3| = 3, the inequality becomes x^2-4(3)<1. Simplifying further, we get x^2-12<1. Moving the constant to the right side, we have x^2<13. Taking the square root, we get x<±√13. Finally, using the UofT Calculus notation, the solution is (-√13, √13), meaning all values of x between -√13 and √13, not including the endpoints.

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