SAT Math Question of the Day

Q: For | x | > 3, which one of the following answers must be true?

1. x > -3
2. x > 3
3. x < -3
4. x² < 9
5. x² > 9

UNANSWERED.

The correct answer is E.

A:

Absolute value means the distance from 0. So, for example, | 4 | is 4, but | -4 | is also 4. On a number line both 4 and -4 are 4 away from 0.

In our problem | x | > 3, if we test out x = 4, we see it works:
| 4 | > 3
   4 > 3

Since we know that x can equal 4, let's try it out in the answers.

A. x > -3
4 > -3   
TRUE

B. x > 3
4 > 3
TRUE

C. x < -3
4 < -3
FALSE, so C can't be the correct answer

D. x² < 9
4 ² < 9
16 < 9
FALSE, so D can't be the correct answer

E. x² > 9
4 ² > 9
16 > 9
TRUE

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

If we test out x = - 4, it's also a valid number for the inequality:
| x | > 3,
| -4 | > 3
   4 > 3

Let's try x = - 4 in the remaining possible answers A, B, and E.

A. x > -3
-4 > -3
FALSE

B. x > 3
-4 > 3
FALSE

E. x² > 9
(-4) ² > 9
16 > 9
TRUE

E is the only possible answer that works for x = 4 or -4.

More detailed explanation:

Above we showed that x = 4 or -4 can work in | x | > 3.

We can try out larger numbers for x like 5, 6, 7, etc, then we see all numbers larger than 3 are valid in | x | > 3. Similarly we can try out numbers smaller than -3 and see that they work too. So far, we know that x > 3 or x < -3.

Let's test a number in the middle of the number line:
If x = 0,
| x | > 3
| 0 | > 3
   0 > 3
So 0 is not a valid number. In fact, all the numbers between -3 and 3 are not valid. 3 and -3 are not valid because the problem | x | > 3 has a > sign not a ≥ . The number line for | x | > 3 looks like the picture below: