# SAT Mathematics : Solving Systems of Inequalities

## Example Questions

### Example Question #1 : Solving Systems Of Inequalities

If  and , which of the following must be true?

Explanation:

Systems of inequalities can be solved just like systems of equations, but with three important caveats:

1) You can only use the Elimination Method, not the Substitution Method. Since you only solve for ranges in inequalities (e.g. a < 5) and not for exact numbers (e.g. a = 5), you can't make a direct number-for-variable substitution.

2) In order to combine inequalities, the inequality signs must be pointed in the same direction.

3) When you're combining inequalities, you should always add, and never subtract.  Since subtraction of inequalities is akin to multiplying by -1 and adding, this causes errors with flipped signs and negated terms. Always look to add inequalities when you attempt to combine them.

With all of that in mind, here you can stack these two inequalities and add them together:

Notice that the  terms cancel, and that with  on top and  on bottom you're left with only one variable, . The new inequality hands you the answer, .

Note that if this were to appear on the calculator-allowed section, you could just graph the inequalities and look for their overlap to use process of elimination on the answer choices. These two inequalities intersect at the point (15, 39).

### Example Question #2 : Solving Systems Of Inequalities

Based on the system of inequalities above, which of the following must be true?

Explanation:

Systems of inequalities can be solved just like systems of equations, but with three important caveats:

1) You can only use the Elimination Method, not the Substitution Method. Since you only solve for ranges in inequalities (e.g. a < 5) and not for exact numbers (e.g. a = 5), you can't make a direct number-for-variable substitution.

2) In order to combine inequalities, the inequality signs must be pointed in the same direction.

3) When you're combining inequalities, you should always add, and never subtract.  Since subtraction of inequalities is akin to multiplying by -1 and adding, this causes errors with flipped signs and negated terms. Always look to add inequalities when you attempt to combine them.

With all of that in mind, you can add these two inequalities together to get:

So . This matches an answer choice, so you're done. Note that process of elimination is hard here, given that  is always a positive variable on the "greater than" side of the inequality, meaning it can be as large as you want it to be. And as long as  is larger than  can be extremely large or extremely small. This is why systems of inequalities problems are best solved through algebra; the possibilities can be endless trying to visualize numbers, but the algebra will help you find the direct, known limits.

### Example Question #1 : Solving Systems Of Inequalities

If x > r and y < s, which of the following must also be true?

x - s > r - y

x - y > r - s

x + y > r + s

xs>ry

x - y > r - s

Explanation:

Because of all the variables here, many students are tempted to pick their own numbers to try to prove or disprove each answer choice. But an important technique for dealing with systems of inequalities involves treating them almost exactly like you would systems of equations, just with three important caveats:

1) You can only use the Elimination Method, not the Substitution Method. Since you only solve for ranges in inequalities (e.g. a < 5) and not for exact numbers (e.g. a = 5), you can't make a direct number-for-variable substitution.

2) In order to combine inequalities, the inequality signs must be pointed in the same direction.

3) When you're combining inequalities, you should always add, and never subtract.  Since subtraction of inequalities is akin to multiplying by -1 and adding, this causes errors with flipped signs and negated terms. Always look to add inequalities when you attempt to combine them.

Here, the first step is to get the signs pointing in the same direction. You already have x > r, so flip the other inequality to get s > y (which is the same thing − you’re not actually manipulating it; if y is less than s, then of course s is greater than y). Now you have:

x > r

s > y

And you can add the inequalities:

x + s > r + y

That’s similar to but not exactly like an answer choice, so now look at the other answer choices. Two of them involve the x and y term on one side and the s and r term on the other, so you can then subtract the same variables (y and s) from each side to arrive at:

x - y > r - s

### Example Question #1 : Solving Systems Of Inequalities

If  and , which of the following must be true?

Explanation:

Systems of inequalities can be solved just like systems of equations, but with three important caveats:

1) You can only use the Elimination Method, not the Substitution Method. Since you only solve for ranges in inequalities (e.g. a < 5) and not for exact numbers (e.g. a = 5), you can't make a direct number-for-variable substitution.

2) In order to combine inequalities, the inequality signs must be pointed in the same direction.

3) When you're combining inequalities, you should always add, and never subtract.  Since subtraction of inequalities is akin to multiplying by -1 and adding, this causes errors with flipped signs and negated terms. Always look to add inequalities when you attempt to combine them.

So what does that mean for you here? You have two inequalities, one dealing with  and one dealing with . But all of your answer choices are one equality with both  and  in the comparison. Since your given inequalities are both "greater than," meaning the signs are pointing in the same direction, you can add those two inequalities together:

Sums to:

And now you can just divide both sides by 3, and you have:

### Example Question #1 : Solving Systems Of Inequalities

Which of the following consists of the -coordinates of all the points that satisfy the system of inequalities above?

Explanation:

Systems of inequalities can be solved just like systems of equations, but with three important caveats:

1) You can only use the Elimination Method, not the Substitution Method. Since you only solve for ranges in inequalities (e.g. a < 5) and not for exact numbers (e.g. a = 5), you can't make a direct number-for-variable substitution.

2) In order to combine inequalities, the inequality signs must be pointed in the same direction.

3) When you're combining inequalities, you should always add, and never subtract.  Since subtraction of inequalities is akin to multiplying by -1 and adding, this causes errors with flipped signs and negated terms. Always look to add inequalities when you attempt to combine them.

Here you have the signs pointing in the same direction, but you don't have the same coefficients for  in order to eliminate it to be left with only  terms (which is your goal, since you're being asked to solve for a range for ). So you will want to multiply the second inequality by 3 so that the coefficients match. That yields:

When you then stack the two inequalities and sum them, you have:

+

Yields:

You can then divide both sides by 4 to get your answer:

### Example Question #6 : Solving Systems Of Inequalities

Which of the following consists of the -coordinates of all of the points that satisfy the system of inequalities above?

Explanation:

This systems of inequalities problem rewards you for creative algebra that allows for the transitive property. You know that , and since you're being asked about  you want to get as much value out of that statement as you can. And while you don't know exactly what  is, the second inequality does tell you about . Note that algebra allows you to add (or subtract) the same thing to both sides of an inequality, so if you want to learn more about , you can just add  to both sides of that second inequality. In doing so,  you'll find that  becomes , or .

Now you have two inequalities that each involve . If  and , then by the transitive property, .

### Example Question #7 : Solving Systems Of Inequalities

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

Explanation:

Systems of inequalities can be solved just like systems of equations, but with three important caveats:

1) You can only use the Elimination Method, not the Substitution Method. Since you only solve for ranges in inequalities (e.g. a < 5) and not for exact numbers (e.g. a = 5), you can't make a direct number-for-variable substitution.

2) In order to combine inequalities, the inequality signs must be pointed in the same direction.

3) When you're combining inequalities, you should always add, and never subtract.  Since subtraction of inequalities is akin to multiplying by -1 and adding, this causes errors with flipped signs and negated terms. Always look to add inequalities when you attempt to combine them.

Here you should see that the  terms have the same coefficient (2), meaning that if you can move them to the same side of their respective inequalities, you'll be able to combine the inequalities and eliminate the  variable. To do so, subtract  from both sides of the second inequality, making the system:

(the first, unchanged inequality)

(the new second inequality)

When you sum these inequalities, you're left with:

Here is where you need to remember an important rule about inequalities: if you multiply or divide by a negative, you must flip the sign. So to divide by -2 to isolate , you will have to flip the sign:

### Example Question #4 : Solving Systems Of Inequalities

If   and , which of the following must be true?

Explanation:

When students face abstract inequality problems, they often pick numbers to test outcomes. But that can be time-consuming and confusing - notice that with so many variables and each given inequality including subtraction, you'd have to consider the possibilities of positive and negative numbers for each, numbers that are close together vs. far apart. There are lots of options.

The more direct way to solve features performing algebra. Notice that with two steps of algebra, you can get both inequalities in the same terms, of . If you add  to both sides of  you get:

And if you add  to both sides of  you get:

If you then combine the inequalities you know that  and , so it must be true that .

### Example Question #2 : Solving Systems Of Inequalities

Which of the following is a possible value of x given the system of inequalities below?

3x+4y > 54

-4x+2y < 10

-3

5

4

-5

5

Explanation:

In order to combine this system of inequalities, we’ll want to get our signs pointing the same direction, so that we’re able to add the inequalities. We’re also trying to solve for the range of x in the inequality, so we’ll want to be able to eliminate our other unknown, y. In order to accomplish both of these tasks in one step, we can multiply both signs of the second inequality by -2, giving us

3x+4y > 54 (our original first inequality)

8x-4y > -10 (our new, manipulated second inequality)

We can now add the inequalities, since our signs are the same direction (and when I start with something larger and add something larger to it, the end result will universally be larger) to arrive at

3x+4y > 54

+8x-4y > -10

11x > 44

Thus, dividing by 11 gets us to

x > 4. Only positive 5 complies with this simplified inequality.

### Example Question #5 : Solving Systems Of Inequalities

Which of the following set of coordinates is within the graphed solution set for the system of inequalities below?

x+2y > 16

-3x+y < 1

(-7, 0)

(1, 7)

(1, 1)

(3, 8)

(3, 8)

Explanation:

In order to combine this system of inequalities, we’ll want to get our signs pointing the same direction, so that we’re able to add the inequalities. We’ll also want to be able to eliminate one of our variables. In order to do so, we can multiply both sides of our second equation by -2, arriving at

x+2y > 16 (our original first inequality)

6x- 2y > -2 (our new, manipulated second inequality)

Adding these inequalities gets us to

x+2y > 16

+6x- 2y > -2

7x > 14

Dividing this inequality by 7 gets us to

x > 2.

Thus, the only possible value for x in the given coordinates is 3, in the coordinate set (3, 8), our correct answer.

Note - if you encounter an example like this one in the calculator-friendly section, you can graph the system of inequalities and see which set applies. The graph will, in this case, look like:

And we can see that the point (3, 8) falls into the overlap of both inequalities. We could also test both inequalities to see if the results comply with the set of numbers, but would likely need to invest more time in such an approach. Here, drawing conclusions on the basis of x is likely the easiest no-calculator way to go!