Q: For | x | > 6, which one of the following answers must be true?
UNANSWERED.
The correct answer is A.
A:
Absolute value means the distance from 0. So, for example, | 7 | is 7, but | -7 | is also 7. On a number line both 7 and -7 are 7 away from 0.
In our problem | x | > 6, if we test out x = 7, we see it works: | 7 | > 6 7 > 6
Since we know that x can equal 7, let's try it out in the answers.
A. x2 > 36 72 > 36 49 > 36 TRUE
B. x3 > 216 73 > 216 343 > 216 TRUE
C. x2 < 36 72 < 36 49 < 36 FALSE, so C can't be the correct answer
D. x3 < 216 73 < 216 343 < 216 FALSE, so D can't be the correct answer
E. x3 < -216 73 < -216 343 < -216 FALSE, so E can't be the correct answer
Since C, D, and E were elimated, only A and B are possible answers.
If we try out x = - 7, it's also a valid number for the inequality: | x | > 6, | -7 | > 6 7 > 6
Let's try x = - 7 in the remaining possible answers A and B.
A. x2 > 36 (-7)2 > 36 49 > 36 TRUE
B. x3 > 216 (-7)3 > 216 -343 > 216 FALSE
A is the only possible answer that works for x = 7 or -7.
More detailed explanation:
Above we showed that x = 7 or -7 can work in | x | > 6.
We can try out larger numbers for x like 8, 9, etc, then we see all numbers larger than 6 are valid in | x | > 6. Similarly we can try out numbers smaller than -6 and see that they work too. So far, we know that x > 6 or x < -6.
Let's test a number in the middle of the number line: If x = 0, | x | > 6 | 0 | > 6 0 > 6 So 0 is not a valid number. In fact, all the numbers between -6 and 6 are not valid. 6 and -6 are not valid because the problem | x | > 6 has a > sign not a ≥ . The number line for | x | > 6 looks like the picture below:
