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GMAT Math Concepts: Inequalities

GMAT Math Concepts: Inequalities

What are inequalities?

Inequalities are some of the more algebraically complex concepts. Inequalities can be defined as statements that show values that are greater or lesser than other values. The open end of the inequality sign(<) will always face the larger value, for example, 5 > 3. Inequalities are used to relate known values as well as variables. For example, you could have a > 3, or x < y.

Inequalities can be manipulated algebraically or combined similarly to equations if all values in the inequality are known to be greater than zero. If you want to manipulate an inequality as you would an equation, you must make sure that all values are positive. This is because if you are multiplying or dividing an inequality by a value less than zero, then you must flip the inequality sign. That is, > becomes < and < becomes >.

Strategic implications

If all values for the inequality are defined as greater than zero, prepare to manipulate algebraically. This means that you must not assume that a variable such as x is positive when it might be negative.  Recognize that inequalities are not the same as equations in considering sufficiency for data sufficiency because there are possibly multiple values. For complex inequalities, you can consider alternative tactics, especially Modeling and Estimation in the problem-solving part of the quantitative section.

Algebraic order of operations

  • The algebraic order of operations applies to inequalities the same way it applies to equations. First, you eliminate fractions and decimals by multiplying the full inequality by the Least Common Denominator(for fractions) and a power of 10 (for decimals).
  • Then work to efficiently simplify the inequality using the four basic functions following the PEMDAS Order, avoiding negatives as much as possible.
  • Lastly, eliminate exponents or radicals by appropriately rooting or raising the full inequality. Remember to account for negatives if need be.

Example

$$\frac{1}{4}(2x^{2}+8)>34$$

  • Multiply the inequality by 4 to eliminate the fraction \(\rightarrow 2x^2+ 8 > 136\)
  • Subtract 8 from each side \( \rightarrow 2x^2> 128\)
  • Divide the inequality by \(2 \rightarrow x^2> 64\)
  • Square root the inequality \(\rightarrow x > 8\)
  • Always remember when square rooting or rooting any even power, that you have to flip the sign to account for the negative possibility \(\rightarrow x < -8\) (Because \(\sqrt{64}=\pm8)\). Furthermore,  x could be less than -8 because if you \(x^2\) is to be greater than 64, then \(-9^{2}=81\) which is greater than 64. The same will apply for all values of x less than -8.

Solving inequalities involving a negative

  • Do as much manipulation as possible first without multiplying or dividing by the negative. Remember that while adding or subtracting inequalities involving negatives, it is not required to flip the inequality sign. 
  • If you have to multiply or divide by a negative at some point, make this the very last thing you do after carrying out all the other manipulations to simplify the process.
  • Finally, if time allows, check the solution against the original inequality to confirm that everything is accurate in terms of the relationship built in the inequality.

Example

$$-\frac{1}{2}(18+y)<-5y$$

  1. Multiply inequality by 2 to get rid of the fraction \(\rightarrow -(18 + y) < -10y\)
  2. Distribute the negative on the left-hand side of the inequality \(\rightarrow-y – 18 < -10y\)
  3. Add y to each side \(\rightarrow-18 < -9y\)
  4. Divide by -9 to isolate y and flip the sign \(\rightarrow2>y\)
  5. Confirm the answer by rearranging the information at step 3: \(\rightarrow-18 < -9y\)

Add 18 and 9y to each side, you are basically just moving the pieces of the inequality from one side to the other side of the inequality \(\rightarrow -18 + 18 + 9y < - 9y + 9y + 18\)

\(\rightarrow 9y < 18\)

Divide the inequality by \(\rightarrow 9 y < 2\).

Solving Compound Inequalities

A compound inequality is inequality with multiple inequality signs. Basically, it has a value in the middle which is greater than one value but less than the other value. e.g a > b > c.

Step 1

Split a single compound inequality into as many simple inequalities as necessary, one for each inequality sign.

Step 2

Seek for the sought value in each individual inequality.

Step 3

Recombine compound inequality if that is helpful in solving the problem, or evaluating data sufficiency.

Example

$$3<\frac{1}{2}(18+y)<-4y$$

  1. split the inequality into two separate inequalities \(\rightarrow 3 < 12 (18 + y)\) and  \(12 (18 + y) < -4y\)
  2. Solve the left inequality first: \(3 < \frac{1}{2} (18 + y)\) 
  • Multiply inequality by \(2\rightarrow 6 < 18 + y\)
  • Subtract 18 from each side \(\rightarrow y > -12\) ( You can keep the y on the right-hand side or take it to the left. It will not make any mathematical difference.) 
  1. Solve the right inequality:  \(\frac{1}{2} (18 + y) < -4y\)
  • Multiply inequality by \(2 \rightarrow18 + y < -8y\)
  • Add 8y to each side \(\rightarrow 18 + 9y< 0\)
  • Subtract 18 from each side \(\rightarrow 9y < -18\)
  • Divide the inequality by \(9\rightarrow y < -2\)
  1. Recombine inequality \(\rightarrow-12 <y < -2\). Y has a range of greater than -12 but less than -2.

Systems of Inequalities

  • Systems of inequalities are used to evaluate individual variables or combinations of variables involving multiple inequalities pending the direction of the inequality signs.
  • Two inequalities with signs facing the same direction can be combined accurately through addition without any change of direction.
  • Two inequalities with signs facing opposite directions can be combined accurately through subtraction taking the direction of the first inequality.

Strategic Implications

Attempt these questions only if you are certain of the steps to complete the combination. It can be a little confusing if you are not very familiar with this kind of manipulation, especially when dealing with subtraction. The addition is much easier, but again, you have to be completely certain if you are going to combine systems under inequalities.

Seek methods of manipulation to facilitate the addition of two inequalities, because it means you can just move variables on either side of each other and get the inequality facing the same direction and be able to add. 

Beware of combining through subtraction because you have to be very technically specific on how to accomplish this.

Adding a System of Inequalities

First, stack the inequalities vertically aligning variables and ensuring the same direction for the inequality signs.

Then carefully add inequalities and make sure to keep the inequality signs facing the same direction.

Finally, follow the standard rules of inequality manipulation to simplify and solve for the individual variables you seek.

Example

Simplify: \(5x +y > 45\) and \(2x – 6 < 2y\)

  1. Stack the inequalities and rearrange signs fo face same direction

\(5x + y > 45\)

\(2y > 2x – 6\)

  1. Add full inequalities \(\rightarrow 5x +3y > 2x + 39\)
  2. subtract 2x from each side \(\rightarrow 3x + 3y > 39\)
  3. Divide inequality by \( 3\rightarrow x+ y > 13\)

Subtracting a System of Inequalities

First stack the inequalities vertically aligning variables and ensure that the signs face opposing directions.

Then carefully subtract the inequalities by applying the sign of the subtracted from inequality(top inequality) for the resulting inequality.

Finally, you can simplify following the rules of inequality manipulation.

Example

What is y greater than if 5x + y > 45 and 2x – 6 < 2y?

  1. Stack the inequalities keeping signs in opposing directions

             \(5x + y > 45\)

             \(2x – 6 < 2y\)

  1. Multiply the top inequality by 2 and the bottom inequality by 5 to produce common factors and eliminate x.

             \(2(5x + y > 45) \rightarrow 10x + 2y > 90\)

             \(5(2x – 6 < 2y)\rightarrow 10x – 30 < 10y\)

  1. Subtract the inequalities applying the top inequality sign

\(2y — 30 > 90 – 10y \rightarrow 2y + 30 > 90 – 10y\)

  1. Add 10y to each side and subtract 30 from each side \(\rightarrow 12y> 60\)
  2. Divide inequality by \(12\rightarrow y > 5\).

Problem  Solving Application

If \(x < -1 < 0 < y < z < 1\), which of the following statements must be true?

  1. \(xy < xz\)
  2. \(xz < 0\)
  3. \(xyz < 0\)

A. I only
B. II only
C. III only
D. I and II
E. II and III

Set up your scratch pad by listing A through E vertically. You can also write down the roman numerals, they could help with elimination as only a limited number are presented for these types of problems. Draw a line on top to write what you seek:

What must be true

A. I  ×

B.  II  ×

C.  III ×

D.  I & II  ×

E.  II & III  ✓

The first thing we need to do is separate the individual relevant variables from the inequality.

   (a) \(x < -1\);   (b) \(0 < y\);     (c) \(y < z\);    (d) \(z < 1\). 

It is not necessary to continue with the second part of the inequality since obviously 0 > -1.

Generally, with algebraic manipulation, even when dealing with inequalities, you may want to try the algebraic manipulation first because that could be more efficient.

For I, \(xy < xz\), we know that x is common to both sides. So we can divide the inequality by \(x\rightarrow xy < xz÷x\). We just have to know if x is positive or negative. We know from (a) that \(x < -1\), which means that the inequality sign is going to flip \(\rightarrow {y} > z\). But we already know from (c) that \(y < z\). This means the Roman numeral I is not true and we can eliminate any answer choice that includes I. That leaves us with choices BCE.

Given the structure of this question, we have to evaluate II and III individually because they are the only options left and both combinations are present in the choices we have left.

We know that x < 0 and z > 0. So no matter the situation, xz has to be less than 0. II will have to be included and that allows us to eliminate anything that does not include it. It leaves us with choices BE.

We also know from this evaluation that xz is negative. If we multiply that by another positive the result will still be negative. xz.y < 0. III is also true. II alone is eliminated and our correct answer is choice E. 

That is the technical approach, we can also take the plugging-in alternative.

What must be true

A. I ×
B. II ×
C. III ×
D. I & II ×
E. II & III ✓

Since we have all the variables x, y, and z in our answer choices, it means we have a plugging-in value opportunity. The best way to do this is to pick easy values for x, y, and z and see what happens.

Let’s take \(x = -2;  y = ¼; z = ½\).

For I, \(xy < xz: xy = -2 × ¼ = -½ ; xz = -2 × ½ = -1\). And we know that \(-1 < -½\). This proves that Roman numeral I is incorrect. Therefore A and D are eliminated.

We then have the same values for x and z.
\(xz < 0, -2 × ½ = -1 < 0\), so our choice must include II.
For \(xyz < 0\), \(y = ¼\) and \(xz = -1\). So xz.\(y = -1 × ¼ = -¼ < 0\). III is also correct and this leads us to choice C as our correct answer.

Data Sufficiency Application

Is \(a + b > 10\)? 

  1. \(2a +3c > 5\)
  2. \(C – 2b < 25\)
  • What we know, \(K: a; b\)  is \(a + b > 10\)? \(y/n\) we can box this off on our whiteboard so that we don’t lose sight of what we are looking for.
  • What we need, N: is one inequality a + b to 10.
  1. \(2a +3c > 5\). This statement doesn’t even have b. On its own, it is not sufficient and that leaves us with options BCE.
  2. \(C – 2b < 25\). This statement on its own will not be sufficient either because it does not have a. We can eliminate choice B.

When we combine the two, we need to subtract to confirm that a + b is not > 10

$$2a+3c>5$$

$$(c-2b<25)3$$

  2a +3c > 5

-6b + 3c < 75
2a + 6b > -70
This statement alone does not definitively tell us whether a + b is or is not greater than 10. We can confidently say that this information is not sufficient and select choice E.

This statement alone does not definitively tell us whether a + b is or is not greater than 10. We can confidently say that this information is not sufficient and select choice E.
The best way to test and improve your own skills as you prepare for the GMAT exam is to go ahead and do as much practice as you can on your own. There are quite a lot of study resources available on our website that you can take advantage of. Get a study package that suits your needs and start preparing for your GMAT exams as soon as today. As you handle real exam questions you’ll gain confidence to sit for your exams. Best wishes.



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