Abs Value Inequality with a Squared Term

In summary: I think I have it now. So if I just solve for x in the second inequality I get ##x < 2## and ##x > -2##, which can be written as ##-2 < x < 2## or in interval notation (-2,2).
  • #1
opus
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Homework Statement


##\left|\left(\frac{x}{2}\right)^2\right| < 1##

Homework Equations

The Attempt at a Solution


The absolute value situation is throwing me off for some reason. Would it be correct to split this into two equations?

##-\left(\frac{x^2}{4}\right) > -1## and ##\frac{x^2}{4} < 1##?
I don't think this is right as the first equation would yield ##x^2 < -4##
 
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  • #2
opus said:
The absolute value situation is throwing me off for some reason.


question
:
can you interchange the absolute value and the squaring? I.e. does

##\left|\left(\frac{x}{2}\right)^2\right| = \left|\left(\frac{x}{2}\right)\right|^2 = \left(\frac{x}{2}\right)^2##

?
 
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  • #3
If ##x## must be real then the problem is trivial, since the square of any real number is positive. So we can drop the original absolute value signs to get ##x^2<4##. What interval on the real number line contains all numbers whose square is less than 4?

It becomes more interesting when ##x## can be complex, so that the ##|\cdot |## signs represent modulus rather than absolute value. Then ##x## is any number inside the circle of radius 2, centred at the origin of the complex plane.
 
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  • #4
opus said:

Homework Statement


##\left|\left(\frac{x}{2}\right)^2\right| < 1##
I suppose the problem statement is missing the fact that you are asked to solve the given inequality. Yes, it is an inequality, not an equation.

Homework Equations



The Attempt at a Solution


The absolute value situation is throwing me off for some reason. Would it be correct to split this into two equations?

##-\left(\frac{x^2}{4}\right) > -1## and ##\frac{x^2}{4} < 1##?
I don't think this is right as the first equation would yield ##x^2 < -4##
You're right, it's not right.

That first inequality, ##\ -\left(\frac{x^2}{4}\right) > -1\,,\ ## simplifies to ##\ \left(\frac{x^2}{4}\right) < 1 \,,\ ## which is the second inequality.

The piece-wise definition of absolute value can be helpful in showing that @StoneTemplePython's simplification is valid.
## |u| =
\begin{cases}
u & \text{if } x \geq 0 \\
-u & \text{if } u < 0
\end{cases} ##

Apply that to ##\left|\left(\frac{x}{2}\right)^2\right| ## and ask yourself if ##\left(\frac{x}{2}\right)^2 ## can be negative.

The real chore is to solve ##x^2 < 4 ## for ##x## .
 
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  • #5
StoneTemplePython said:

question
:
can you interchange the absolute value and the squaring? I.e. does

##\left|\left(\frac{x}{2}\right)^2\right| = \left|\left(\frac{x}{2}\right)\right|^2 = \left(\frac{x}{2}\right)^2##

?
I want to say that if we have an even power, then yes we can.
andrewkirk said:
If ##x## must be real then the problem is trivial, since the square of any real number is positive. So we can drop the original absolute value signs to get ##x^2<4##. What interval on the real number line contains all numbers whose square is less than 4?

It becomes more interesting when ##x## can be complex, so that the ##|\cdot |## signs represent modulus rather than absolute value. Then ##x## is any number inside the circle of radius 2, centred at the origin of the complex plane.
(-2,2) would be the answer then. And your second part sounds interesting.
SammyS said:
I suppose the problem statement is missing the fact that you are asked to solve the given inequality. Yes, it is an inequality, not an equation.

You're right, it's not right.

That first inequality, ##\ -\left(\frac{x^2}{4}\right) > -1\,,\ ## simplifies to ##\ \left(\frac{x^2}{4}\right) < 1 \,,\ ## which is the second inequality.

The piece-wise definition of absolute value can be helpful in showing that @StoneTemplePython's simplification is valid.
## |u| =
\begin{cases}
u & \text{if } x \geq 0 \\
-u & \text{if } u < 0
\end{cases} ##

Apply that to ##\left|\left(\frac{x}{2}\right)^2\right| ## and ask yourself if ##\left(\frac{x}{2}\right)^2 ## can be negative.

The real chore is to solve ##x^2 < 4 ## for ##x## .
Ah that piecewise comment helps. Thanks!
 
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1. What is an absolute value inequality with a squared term?

An absolute value inequality with a squared term is an inequality that contains an absolute value expression and a squared term (x^2) on one side of the inequality symbol. It can be written in the form |ax^2 + bx + c| < d or |ax^2 + bx + c| > d, where a, b, c, and d are real numbers.

2. How do you solve an absolute value inequality with a squared term?

To solve an absolute value inequality with a squared term, you need to first isolate the absolute value expression on one side of the inequality symbol. Then, you can solve for the variable by taking the square root of both sides of the inequality. Remember to consider both the positive and negative solutions when solving for the variable.

3. What is the difference between solving an absolute value inequality with a squared term and a linear absolute value inequality?

The main difference between solving an absolute value inequality with a squared term and a linear absolute value inequality is that when solving a squared term inequality, you need to consider both the positive and negative solutions. This is because taking the square root of both sides of the inequality can result in two solutions. However, when solving a linear absolute value inequality, there is only one solution to consider.

4. Can you graph an absolute value inequality with a squared term?

Yes, you can graph an absolute value inequality with a squared term. The graph will be in the shape of a parabola and will have a "V" or "U" shape depending on whether the inequality symbol is < or >. You can use the vertex and the x-intercepts of the parabola to determine the solution to the inequality.

5. What are some real-life applications of absolute value inequalities with a squared term?

Absolute value inequalities with a squared term can be used in real-life situations such as determining the maximum or minimum values of a function, finding the range of possible values for a variable, or solving optimization problems. They can also be used in physics and engineering to model and solve problems related to motion and distance.

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