# SAT Mathematics : Understanding Inequalities

## Example Questions

### Example Question #32 : Inequalities & Absolute Value

Which of the following expresses the complete set of values for  that satisfy the inequality above?

Explanation:

Inequalities can be solved just like equations, with one important caveat: if you multiply or divide by a negative number, you have to flip the inequality sign. Here, as you will see, there is no need to do that, so you can solve this just like you would an equation. First, multiply both sides by 2 to eliminate the denominator:

Next, add 2 to both sides to isolate the  term:

Then divide both sides by 3 to get  alone:

### Example Question #1 : Understanding Inequalities

Which of the following represents the complete set of values of  that satisfy the inequality above?

Explanation:

This problem showcases an important concept about working with inequalities. If you multiply or divide an inequality by a negative number, you must flip the inequality sign. Here you might divide both sides of the original inequality by  to get  alone. If you do so, however, you must flip the sign to get:

You may wish to choose a number consistent with your answer to make sure that it satisfies the original inequality. If you were to choose  you would see that the given inequality becomes: ,  which works because

Note that on many problems it is possible to avoid multiplying/dividing by a negative by adding or subtracting (which have no such restrictions) terms to sides of the inequality where they will be positive. For example, given , if you add  to both sides and subtract 10 from both sides, you'll end up with , and now you will divide by positive 2, and not have to worry about sign-flipping to arrive at .

### Example Question #1 : Understanding Inequalities

If , then which of the following must be true?

I.

II.

III.

I and III only

II and III only

I only

None of these must be true

None of these must be true

Explanation:

At first glance, you might simply multiply both sides of the original inequality by  and believe that . However, what if  is a negative number? Remember that when you multiply or divide by a negative number in an inequality, you must flip the sign. But here you don't know whether multiplying by  would mean multiplying by a positive number or a negative number, so you don't know which direction the sign should point. An important consideration in manipulating inequalities is that you can never multiply or divide by a variable unless you know the sign of the variable. As a result both I and II are possible but neither is guaranteed:

1. If  is positive then

2. If  is negative then you must flip the sign and

3. The given statement implies that x and y are either both negative or both positive but you do not know which is the case.

As a result, none of these statements “must be true” so the correct answer is "none of these must be true."

### Example Question #4 : Understanding Inequalities

If , which of the following must be true?

Explanation:

Inequalities can be solved just like equations (with the caveat that if you multiply/divide by a negative you must flip the sign of the inequality, but that does not come in to play on this problem). So you can solve this by doing the same thing to both sides of the inequality. Starting with , you can:

1) Add 2 to both sides to get:

2) Divide both sides by 2 to get:

### Example Question #2 : Understanding Inequalities

Which of the following represents the full set of values of  that satisfy the above inequality?

Explanation:

Inequalities can be solved just like equations can, with one important caveat as you'll see in this problem: if you multiply or divide by a negative number, you have to flip the direction of the inequality sign.

Here you can start by multiplying both sides by 2 in order to eliminate the denominator. That brings you to:

Now you have a choice for how you move the remaining terms to isolate and solve for the variable. If you subtract 6 from both sides, you'll have:

Here note that in order to solve for the solution set of , you would have to divide by -1 to remove the negative. If you do so, you'll need to flip the direction of the sign to get the proper answer:

Now, also recognize that you could have avoided the divide-by-negative step altogether.  At this point:

You could have added  to both sides to get:

And then when you subtract 8 from both sides you also end up at the correct answer:

### Example Question #6 : Understanding Inequalities

Which of the following represents the complete set of values of  that satisfy the inequality above?

Explanation:

Inequalities can be solved just like equations, with the only caveat that if you multiply or divide by a negative number you have to reverse the sign of the inequality. In this problem, you should not have to take that step.

First here, your goal is to get all the  terms on one side and the numeric terms on the other. To do so, you can add  to both sides and subtract  from both sides to get:

Now you want to get  alone, so you can divide both sides by . This leaves:

, or the reduced fraction .

Note that you can read inequalities from left to right or from right to left, so here "two thirds is greater than " is the same thing as " is less than two-thirds." So you will find the answer written as .

Note that you might want to test a value close to your inequality to make sure that it works. Since you have , you might try . If you plug that in to the given inequality you'll see that it does work, returning the true statement . This helps confirm your answer.

### Example Question #3 : Understanding Inequalities

Which of the following represents the complete solution set for values of  that satisfy the inequality above?

Explanation:

A very important consideration when working with inequalities is that whenever you multiply or divide by a negative number, you must flip the inequality sign. Here that means that if you take the (logical) first step of dividing both sides by , you need to flip the inequality sign from greater-than to less-than, to get:

Note that you could also handle this by only using addition/subtraction where the rules are exactly the same as with equations. To do that, add  and subtract  from both sides of the given inequality.  then becomes:

Then you're dividing by a positive number, , to isolate . This gives you , matching the right answer determined above.

### Example Question #4 : Understanding Inequalities

Which of the following inequalities represents the complete solution set for values of  that satisfy the inequality above?

Explanation:

Inequalities can be solved the same way as equations - do the same thing to both sides until you've isolated the variable - with one important exception. If you multiply or divide by a negative number, you must reverse the inequality sign. That situation (as you'll see here) is typically easy to avoid.

Here if you subtract  from both sides, you can get all the  terms on one side of the equation, and with a positive coefficient so you do not need to worry about dividing by a negative. That gives you:

Then add  to both sides to isolate the variable on the right:

, which is the same as

### Example Question #8 : Understanding Inequalities

Which of the following expresses the complete set of values for  that satisfy the inequality above?

Explanation:

In order to tackle this question, we'll want to keep in mind that inequalities follow all the same rules and operations as equations - with one important exception. When we multiply or divide by a negative, we have to flip the inequality sign, as we are basically "flipping" the relationship over 0 on the number line. (A number that is larger when positive becomes "more negative," and thus smaller when negative, and vice versa).

In this case, we start by getting rid of our denominator by multiplying both sides of our inequality by 4

we can then subtract two from both sides to get to

and divide both sides by -3, (keeping in mind that we need to flip the sign when we do so!)