Absolute Value Inequalities
Some GRE problems present inequalities involving absolute value where there is an absolute value expression on one side and a number on the other side. In other words, the inequality has the form
|Something| < Number
OR
|Something| > Number.
To understand how this works, take a look at the following problem:
If
$$\displaystyle |x -8| < 5$$,
what is the range of values for _x_?
How would you approach this problem?
Incorrect.
Excellent!
This is the right approach.
Notice that the -8 is inside the absolute value brackets, and you can't just move it to the other side. First, you need to get rid of the absolute value brackets. The absolute value is never negative. However, the expression inside it may be either positive or negative. So you need to consider both situations.
With this in mind, what are the two possible scenarios?
Incorrect.
Great!
In the first scenario, _x_ - 8 is positive and
$$\displaystyle |x - 8| = x - 8$$.
The inequality becomes
$$\displaystyle x - 8 < 5$$.
In the second scenario, _x_ - 8 is negative and
$$\displaystyle |x -8| = -(x - 8)$$.
The inequality becomes
$$\displaystyle -(x - 8) < 5$$.
The inequality
$$\displaystyle -(x - 8) < 5$$
from the second scenario needs to be simplified. How would you simplify this inequality?
Not quite. This is not incorrect, but the inequality
$$\displaystyle -x + 8 < 5$$
could be much simpler.
Great!
Multiplying the inequality by -1 gets rid of the parentheses and simplifies the inequality greatly.
Which one of the following would you say is the result of multiplying the inequality
$$\displaystyle -(x - 8) < 5$$
by -1?
Incorrect.
Correct!
When multiplied by a negative number, an inequality flips, and in this case the "<" becomes ">."
In fact, you can save time from the beginning by writing the second scenario as
$$\displaystyle x - 8 > -5$$ instead of
$$\displaystyle -(x - 8) < 5$$.
You only need to remember to flip the inequality and switch the sign of the number on the right-hand side of the inequality.
The two possible scenarios are
$$\displaystyle x - 8 < 5$$
and
$$\displaystyle x - 8 > -5$$.
How would you solve the two inequalities?
Not exactly. This isn't wrong. But you need to be careful because the -8 from the left-hand side of the inequality arrives as 8 on the right-hand side. You should try to avoid situations that provide opportunities for mistakes.
Correct. This is the right move for simplifying these inequalities.
By adding 8 to both sides of the inequalities in the two scenarios you obtain
$$x < 13$$
and
$$\displaystyle x > 3$$.
Finally, you can combine these two inequalities into one:
$$\displaystyle 3 < x < 13$$.
To sum this up:
[[summary]]
I would move the -8 to the other side.
I would remove the absolute value brackets.
$$x - 8 < 5$$
and
$$\displaystyle x - 8 < -5$$
$$x - 8 < 5$$
and
$$\displaystyle -(x - 8) < 5$$
I would get rid of the parentheses.
I would multiply the inequality by -1.
$$x - 8 < -5$$
$$x - 8 > -5$$
I would move the -8 to the other side.
I would add 8 to both sides.
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