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December 09, 2019, 10:03:41 am

Author Topic: Absolute Value Inequatily  (Read 189 times)  Share 

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Jefferson

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Absolute Value Inequatily
« on: August 14, 2019, 10:48:47 pm »
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Hi, 
When solving the inequation

| x + 4 | + | x - 3 |  ≥ 7 using algebra,

I get 3 cases.

Case 1: for x < 4
Solving gives x ≤ 4

Case 2: for -4 ≤ x < 3
Solving gives 7 ≥ 7 (gradient of 1 cancels out, leaving horizontal line)

Case 3: for x ≥ 3
Solving gives x ≥ 3

Therefore, I concluded based on the algebra (with the help of Desmos, which shaded only the following regions) that:
x ≤ 4   OR   x ≥ 3

However, it's obvious that this is true for any x values, evident in the graph itself. The section -4 ≤ x ≤ 3 will be 7 for all values of x, which is something the algebra couldn't show.
How should this result be interpreted in the exam? What explanation can we offer?
Would the answer be "True for all real x"?

Desmos Graph
https://www.desmos.com/calculator/frdjmru5xl

Thank you.
« Last Edit: August 14, 2019, 10:56:04 pm by Jefferson »

blyatman

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Re: Absolute Value Inequatily
« Reply #1 on: August 14, 2019, 11:15:39 pm »
+1
Hi, 
When solving the inequation

| x + 4 | + | x - 3 |  ≥ 7 using algebra,

I get 3 cases.

Case 1: for x < 4
Solving gives x ≤ 4

Case 2: for -4 ≤ x < 3
Solving gives 7 ≥ 7 (gradient of 1 cancels out, leaving horizontal line)

Case 3: for x ≥ 3
Solving gives x ≥ 3

Therefore, I concluded based on the algebra (with the help of Desmos, which shaded only the following regions) that:
x ≤ 4   OR   x ≥ 3

However, it's obvious that this is true for any x values, evident in the graph itself. The section -4 ≤ x ≤ 3 will be 7 for all values of x, which is something the algebra couldn't show.
How should this result be interpreted in the exam? What explanation can we offer?
Would the answer be "True for all real x"?

Desmos Graph
https://www.desmos.com/calculator/frdjmru5xl

Thank you.

Your calculations for case 2 are correct, but your final interpretion of your result is incorrect.

You made an initial assumption: -4<x<3, and from this, you reached the conclusion that 7>=7.  This statement is ALWAYS true. If you backwork your solution, this means that your previous line must always be true, which means that the line before is also always true, and so on, all the way down to the first line of working: i.e. the very first statement is also ALWAYS true. Hence, |x+4| + |x-3|  is always >=7 for -4<x<3. I.e. the entire domain -4<x<3 satisfies that inequality.
« Last Edit: August 14, 2019, 11:40:04 pm by blyatman »
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Research background: General Relativity (Gravitational Astrophysics Research Group, Sydney Institute for Astronomy, USYD)
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