SAT Math Question of the Day

Q: For | x | > 8, which one of the following answers must be true?

1. x > -8
2. x > 8
3. x < -8
4. x² < 64
5. x² > 64

UNANSWERED.

The correct answer is E.

A:

Absolute value means the distance from 0. So, for example, | 10 | is 10, but | -10 | is also 10. On a number line both 10 and -10 are 10 away from 0.

In our problem | x | > 8, if we test out x = 10, we see it works:
| 10 | > 8
   10 > 8

Since we know that x can equal 10, let's try it out in the answers.

A. x > -8
10 > -8   
TRUE

B. x > 8
10 > 8
TRUE

C. x < -8
10 < -8
FALSE, so C can't be the correct answer

D. x² < 64
10 ² < 64
100 < 64
FALSE, so D can't be the correct answer

E. x² > 64
10 ² > 64
100 > 64
TRUE

Since C and D were elimated, only A, B, and E are possible answers.

If we test out x = -10, it's also a valid number for the inequality:
| x | > 8,
| -10 | > 8
   10 > 8

Let's try x = -10 in the remaining possible answers A, B, and E.

A. x > -8
-10 > -8
FALSE

B. x > 8
-10 > 8
FALSE

E. x² > 64
(-10) ² > 64
100 > 64
TRUE

E is the only possible answer that works for x = 10 or -10.

More detailed explanation:

Above we showed that x = 10 or -10 can work in | x | > 8.

We can try out larger numbers for x like 9, 11, etc, then we see all numbers larger than 8 are valid in | x | > 8. Similarly we can try out numbers smaller than -8 and see that they work too. So far, we know that x > 8 or x < -8.

Let's test a number in the middle of the number line:
If x = 0,
| x | > 8
| 0 | > 8
   0 > 8
So 0 is not a valid number. In fact, all the numbers between -8 and 8 are not valid. 8 and -8 are not valid because the problem | x | > 8 has a > sign not a ≥ . The number line for | x | > 8 looks like the picture below: