Q: For | x | < 9, which one of the following is the best answer?
UNANSWERED.
The correct answer is A.
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 | < 9, if we test out x = 3, we see it works: | 3 | < 9 3 < 9
Since we know that x can equal 3, let's try it out in the answers.
A. x2 < 81 32 < 81 9 < 81 TRUE
B. x2 > 81 32 > 81 9 > 81 FALSE, so B can't be the correct answer
C. x3 > 729 33 > 729 27 > 729 FALSE, so C can't be the correct answer
D. x3 < -729 33 < -729 27 < -729 FALSE, so D can't be the correct answer
E. x < 9 3 < 9 TRUE
Since B, C and D were elimated, only A and E are possible answers.
Let's try a negative number: x = -2. | x | < 9 | -2 | < 9 2 < 9 It works!
Let's try x = -2 in the remaining possible answers A and E.
A. x2 < 81 (-2)2 < 81 4 < 81 TRUE
E. x < 9 -2 < 9 TRUE
Either A or E still could be a possible answer.
What happens if we choose x = -10?
| x | < 9 | -10 | < 9 10 < 9 So x can not equal -10.
What if we plug -10 into the possible answers A or E?
E. x < 9 -10 < 9 TRUE This is not good. We just said that x can not equal -10, yet it's TRUE when we plug it into this inequality. So x < 9 includes EXTRA values for x, such as -10, -11, -12, etc, that don't meet our original requirement of | x | < 9. So this is not the best answer.
A. x2 < 81 (-10)2 < 81 100 < 81 FALSE We said that x can not equal -10, and sure enough when we plug it into x2 < 81 we get a FALSE result. That makes sense. A is the best answer.
More detailed explanation:
Above we showed that x = 3 or -2 can work in | x | < 9.
We can try out other numbers for x like -4, 0, 1, etc, then we see all numbers in this middle range between -9 and 9 are valid in | x | < 9. For example, if x = 0, | x | < 9 | 0 | < 9 0 < 9
Now we can try out numbers greater than 9 and see that they fail. If x = 10, | x | < 9 | 10 | < 9 10 < 9
Similarly try numbers less than -9 and they also fail. If x = -12, | x | < 9 | -12 | < 9 12 < 9
So 10 and -12 are not a valid numbers. In fact, all the numbers less than -9 or greater than 9 are not valid. 9 and -9 are not valid because the problem | x | < 9 has a < sign not a ≤ . The number line for | x | < 9 looks like the picture below:
