# Inequalities:

The best way to think of inequalities is as equations with slightly less specific information. Instead of providing an exact location, they provide a range of locations, all of which are satisfactory. With “=” instead of“<”, we can solve for the exact number. In an inequality, that number will be used as an endpoint.

**Find the Range**

To find the range, you can solve for each inequality as if it were an equation, and then look at the direction of the sign. For example:

To find the range of values for x, we can subtract 3 from all three expressions, and then divide by 2.

If the question had asked if x³ < 0 (if x is negative), then we would not have definitive proof. Inequalities literally provide a range of options, so be sure what critical points (0, +1, -1 etc) these ranges include.

**How many Ranges? It Depends.**

When you have one greater/less than sign in an inequality with a linear variable, there is only one range created. Here are your options:

x>2 OR

x<4

These both provide one range stretching to infinity or to negative infinity. But what if you had x² < 9?

x² < 9 actually provides a finite range with two endpoints, even though there is only one < sign. The endpoints in this case are +/-3. Any x value between these two points will satisfy the equation. Also note that the expression x² < 9 can also be written as |x| < 3.

Be careful how many endpoints are created, to recognize whether the range is finite or infinite.

** Multiplication/Division of a Negative Number**

Make sure you are not multiplying or dividing by a negative number. In these situations you must FLIP the signs. If you aren’t sure whether the number is positive or negative (as in a variable) then you cannot perform the action. Please see the following Data Sufficiency question:

If m ≠ -n, is (m – n)/(m + n) > 1?

(1) n < -2

(2) m > 1

On first look, this seems pretty simple. We always want to make the inequality as simple as possible before analyzing, so we can multiply both sides by (m + n) to get (m – n) > (m + n). Then subtract m from both sides to get –n < n. If n<0, as provided in Statement (1), then there is sufficient information, right?

Not so fast. How were we so sure that (m + n) was positive? If it were negative, then we would have to flip the sign, which would yield an opposite answer. Even with both (1) and (2), we still do not know the sign of (m + n). We may know that the numerator (m – n) is positive, but since no specific magnitude is given for m or n, we cannot know the sign of their difference. If given absolute equations for m and n, we would be able to solve. Choice E.

**Inequalities and Absolute Values**

Since an absolute value simply expresses a distance from a certain point, if we used an inequality instead of an equal sign, we know if a distance is more or less than the specified amount. Take a look at the following example:

Which inequality below most accurately represents the range of possible values for x?

A. The absolute value of x is less than or equal to 4.

B. The absolute value of x is less than or equal to 5.

C. The absolute value of (x + 2) is less than or equal to 2.

D. The absolute value of (x – 1) is less than or equal to 3.

E. The absolute value of (x + 1) is less than or equal to 3.

The first thing we want to do is determine the midpoint between the endpoints. What this does is create an equal distance in the negative and positive directions from that center location. For this question, the center point between -2 and +4 is +1.

From +1, our range of possible values for x extends at most 3 in either direction, but can also be 1, 1.5, 2.8 etc in either direction. This means that the distance from +1 is LESS THAN 3.

Looking at our answer choices, we now have to decide between D and E. With absolute values, as with some advanced pre-calculus equations, the shift within the equation is opposite of the shift on the graph. A good way to quickly test this is to plug in numbers toward the extremes and see which fit.

For (E), if we plug in x = 3.5 (which we know is within the given range) into | x + 1 | ≤ 3, we get 4.5 ≤ 3, which is false.

Any x within the given range meets the inequality | x – 1 | ≤ 3 in Choice D. Be sure to think of ranges in terms of both inequalities and absolute values, as these come up in about 1-2 questions per test.

Good luck. As always, please post comments below, and add any other tips for future readers!

## 6 comments

H Young on November 2nd, 2009 at 8:57 pm

Why is it not choice A for your second question above? The absolute value of x is less than or equal to 4, which is true if you plug in all the possible numbers on the number line. In fact, choice B, the absolute value of x is less than or equal to 5, is correct as well. Please explain why the correct answer is choice D instead.

Thanks.

Jake Becker on November 2nd, 2009 at 9:07 pm

While you are correct that each point on the line fits the descriptions provided in A and B, they are not exact. There are points in the range provided in A (like -3) that fit its description but are not in the image provided. So while it fits, it is overinclusive.

D represents the exact range, and does not include anything not in the range.

Keep an eye out for phrases like "most accurately" in the stimulus.

pink on November 4th, 2009 at 3:29 pm

In the problem, m!=n , (m-n)/(m+n) > 1 ?

1)n1

My doubt is - why can't it be A). After simplifying the question, we get -n< n. So i plug in different values for n.

( -3, 0, 3) So the values -3 and 0 violates the equation( -n < n). And only the +ve value i.e +3 satifies (-n 1 , there is no variable in the equation (-n < n). So we can eliminate B,C,D,E

pink on November 4th, 2009 at 3:31 pm

Posting again, as some characters were fudged

In the problem, m!=n , (m-n)/(m+n) > 1 ?

1) n 1

My doubt is - why can't it be A). After simplifying the question, we get -n < n. So i plug in different values for n.

( -3, 0, 3) So the values -3 and 0 violates the equation

( -n < n). And only the +ve value i.e +3 satifies (-n 1 , there is no variable in the equation (-n < n). So we can eliminate B,C,D,E

Reply to this comment

Jake Becker on November 4th, 2009 at 5:20 pm

Pink,

When you multiply both sides by (M + N) to then get -n 1, which is impossible. But even so, (1) tells you that n<-2, so n=0 is not an option regardless.

Hesham on November 10th, 2009 at 9:21 pm

1) HOw is this so in the first problem:

We may know that the numerator (m – n) is positive ?

2) Does exercise 2 relevant to questions exactly similar to it, wiht number line shown, or we can draw a line to help us solve many of those crazy inequalities and absolute values ? Please, provide example.