Q: For | x | < 4, which one of the following answers is the best answer?
UNANSWERED.
The correct answer is C.
A:
Absolute value means the distance from 0. So, for example, | 3 | is 3, but | -3 | is also 3. On a number line both 3 and -3 are 3 away from 0.
In our problem | x | < 4, if we test out x = 3, we see it works: | 3 | < 4 3 < 4
Since we know that x can equal 3, let's try it out in the answers.
A. x < 4 3 < 4 TRUE
B. x > 4 3 > 4 FALSE, so B can't be the correct answer
C. x2 < 16 32 < 16 9 < 16 TRUE
D. x2 > 16 32 > 16 9 > 16 FALSE, so D can't be the correct answer
E. x3 > 64 33 > 64 27 > 64 FALSE, so E can't be the correct answer
Since B, D and E were elimated, only A and C are possible answers.
If we test out x = -3, it's also a valid number for the inequality: | x | < 4 | -3 | < 4 3 < 4
Let's try x = -3 in the remaining possible answers A and C.
A. x < 4 -3 < 4 TRUE
C. x2 < 16 (-3) 2 < 16 9 < 16 TRUE
Either A or C still could be a possible answer.
What happens if we choose x = -5?
| x | < 4 | -5 | < 4 5 < 4 So x can not equal -5.
What if we plug -5 into the possible answers A or C?
A. x < 4 -5 < 4 TRUE This is not good. We just said that x can not equal -5, yet it's TRUE when we plug it into this inequality. So x < 4 includes EXTRA values for x, such as -5, -6, -7, etc, that don't meet our original requirement of | x | < 4. So this is not the best answer.
C. x2 < 16 (-5) 2 < 16 25 < 16 FALSE We said that x can not equal -5, and sure enough when we plug it into x2 < 16 we get a FALSE result. That makes sense. C is the best answer.
More detailed explanation:
Above we showed that x = 3 or -3 can work in | x | < 4.
We can try out other numbers for x like -2, 0, 1, etc, then we see all numbers in this middle range between -4 and 4 are valid in | x | < 4. For example, if x = 0, | x | < 4 | 0 | < 4 0 < 4
Now we can try out numbers greater than 4 and see that they fail. If x = 5, | x | < 4 | 5 | < 4 5 < 4
Similarly try numbers less than -4 and they also fail. If x = -5, | x | < 4 | -5 | < 4 5 < 4
So 5 and -5 are not a valid numbers. In fact, all the numbers less than -4 or greater than 4 are not valid. 4 and -4 are not valid because the problem | x | < 4 has a < sign not a ≤ . The number line for | x | < 4 looks like the picture below:
