6. Inequalities & Absolute Value

Question 1

What does it mean to solve an inequality?

Answer 1

The same thing it means for an equation: find the value or values of x that make the inequality true. When you plug a solution back into the original equation or inequality, you get a true statement.

The difference is that equations have only one (or just a few) values as solutions. In contrast, inequalities give a whole range of values as solutions – way too many to list individually

Question 2

How do you solve an inequality? (e.g. 45 < -5w)

Answer 2

Isolate variable by transforming each side. You are always allowed to simplify an expression on just one side of an inequality as long as you do not change the expression’s value

All of the Golden Rule moves apply (e.g. adding, subtracting, dividing, multiplying) except when you multiply or divide both sides of an inequality by a negative number, in which case you must flip the inequality sign

Example:
45 < -5w
-9 > w

Question 3

What do you do if you have to multiply or divide both sides of an inequality by a negative number?

Answer 3

Flip the inequality sign

Question 4

Can you cross multiply variables across an inequality?

Answer 4

Usually you cannot cross multiply variables across an inequality because those variables might be negative. However, if you are told that all variables are positive, then it is ok to cross multiply

Question 5

How do you combine inequalities?

Answer 5

Converting several inequalities to a compound inequality, which is a series of inequalities strung together

(1) Solve any inequalities that need to be solved
(2) Simplify the inequalities so that all the inequality symbols point in the same direction, preferably to the left
(3) Line up the common variables in the inequalities

Question 6

If x > 8, x < 17, and x + 5 < 19, what is the range of possible values for x?

Answer 6

x + 5 < 19
x < 14

8 < x
x < 17
x < 14

8 < x < 14

Note:

• x < 14 is more limiting than x < 17 (i.e. whenever x < 14, x will always be less than 17, but not vice versa). Discard the less limiting inequality
• It is not always possible to combine all the inequalities

Question 7

How do you manipulate compound inequalities?

Answer 7

Must perform operations on every term in the inequality, not just the outside terms
Example 1:
x + 3 < y < x + 5
x < y – 3 < x + 2

Example 2:
c/2 < b – 3 < d/2
c < 2b – 6 < d

Question 8

How do you add inequalities together?

Answer 8

(1) Line up the inequalities so that they are all facing the same direction
(2) Take the sum of the two inequalities

Note:

• NEVER subtract or divide two inequalities
• Only multiply inequalities together under certain circumstances (i.e. if both sides of both inequalities are positive)

Question 9

What is an Absolute Value?

Answer 9

Absolute value of a number describes how far that number is away from 0. It is the distance between that number and 0 on a number line

NOTE: Treat the absolute value symbols likes parentheses (i.e. solve the arithmetic problem inside first, and then find the absolute value of the answer

Question 10

What do you do with a variable inside absolute value signs? e.g. ǀyǀ = 3)

Answer 10

This equation has two solutions. There are two numbers that are three units away from 0: namely 3 and -3. So both of these numbers could be possible values for y. y is either 3 or -3

Question 11

How do you find the solution for an absolute value equation? e.g. 6 * ǀ2x + 4ǀ = 30

Answer 11

Step 1: isolate the absolute value expression on one side of the equation
6 * ǀ2x + 4ǀ = 30
ǀ2x + 4ǀ = 5

Step 2: drop the absolute signs and set up two equations. The first equation has the positive value of what’s inside the absolute value. The second equation puts in a negative sign
+(2x + 4) = 5
OR
-(2x + 4) = 5

Step 3: solve both equations/inequalities and check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides are equal
x = 1/2
x = -9/2

*NOTE: it is possible for one of the solutions to fail!

Question 12

What happens if your arrows overlap on the number line?

Answer 12

Only the numbers that fall within the range of both arrows will be solutions to the inequality

Example:
ǀy + 3ǀ < 5
y + 3 < 5
y < 2

OR

-y – 3 < 5
Y > - 8

Only the numbers between -8 and 2 represent possible solutions

Question 13

How should you rephrase an inequality with an absolute value expression? e.g. is |n| < 4?

Answer 13

Open up the absolute value signs. There are two scenarios for the inequality |n| < 4

If n > 0, the question becomes, “Is n < 4?”
If n < 0, the question becomes, “Is n > -4?”

We can then combine the two questions: “Is -4 < n < 4?”

Question 14

How do you solve an equation that contains one variable and at least one constant in more than one absolute value expression? e.g. ǀx – 2ǀ = ǀ2x – 3ǀ

Answer 14

Only consider two cases: one in which neither expression changes sign (positive/positive), and another in which one expression changes sign (positive/negative)

Case A (Same Sign):
x – 2 = 2x – 3
x = 1
Validity Check: ǀ1 – 2ǀ = ǀ2 – 3ǀ = 1

Case B (Different Signs):
x – 2 = -2x + 3
x = 5/3
Validity Check: ǀ5/3 – 6/3ǀ = ǀ10/3 – 9/3ǀ = 1/3

Question 15

How do you solve algebra problems with integer constraints?

Answer 15

Simplify the equation and back solve

Question 16

How do you solve an inequality with a square root expression? e.g. n^2 > 16

Answer 16

If n > 0, then n > 4
OR
If n < 0, then n < -4

*NOTE: YOU CAN ONLY TAKE THE SQUARE ROOT OF AN INEQUALITY FOR WHICH BOTH SIDES ARE DEFINITELY NOT NEGATIVE, SINCE YOU CANNOT TAKE THE SQUARE ROOT OF A NEGATIVE NUMBER

Question 17

What is the formula for interpreting absolute value problems?

Answer 17

ǀx + bǀ = c

• The center point of the graph is –b
• The equation tells us that x must be exactly c units away from -b

ǀx + bǀ < c

• The center point of the graph is –b
• The less than symbol tells us that x must be less than c units away from -b

Question 18

If x^2 < 4, what are the possible values for x?

Answer 18

Sqrt(x^2) < Sqrt(4)
ǀxǀ < 2

If x is positive, then x < 2
If x is negative, then x > 2

Question 19

Ch 8, Q 22. Solve the following inequality (x/3) + 8 < (x/2)

Answer 19

(x/3) + 8 < (x/2)
= 2x + 48 < 3x Multiply both sides by 6
= 48 < x

Question 20

Ch 8, Q 23. Solve the following inequality 2x – 1.5 > 7

Answer 20

2x – 1.5 > 7
2x > 8.5
x > 4.25

Question 21

Ch 8, Q 38. Solve the following equation ǀ3x - 4ǀ = 2x + 6

Answer 21

ǀ3x - 4ǀ = 2x + 6
3x – 4 = 2x + 6
x = 10

OR

-3x + 4 = 2x + 6
4 = 5x + 6
-2 = 5x
x = -2/5

Question 22

Ch 8, Q 39. Solve the following equation ǀ5x - 17ǀ = 3x + 7

Answer 22

ǀ5x - 17ǀ = 3x + 7
5x - 17 = 3x + 7
2x = 24
x = 12

OR

-5x + 17 = 3x + 7
10 = 8x
x = 10/8 = 5/4

Question 23

Ch 8, Q 47. Solve the following inequality ǀx + 4ǀ/2 > 5

Answer 23

ǀx + 4ǀ/2 > 5
x + 4 > 10
x > 6

OR

-x - 4 > 10
x < -14

Question 24

Ch 8, Q 55. Solve the following inequality 3ǀ(5x/3) - 7ǀ < (2x/3) + 18

Answer 24

3ǀ(5x/3) - 7ǀ < (2x/3) + 18
(5x/3) – 7 < 2x/9 + 6
(5x/3) < 2x/9 + 13
(15x/9) – (2x/9) < 13
13x/9 < 13
13x < 117
x < 117/13
x < 9
OR

(-5x/3) + 7 < 2x/9 + 6
(-5x/3) < 2x/9 - 1
(-15x/9) < 2x/9 - 1
(-15x/9) – (2x/9) < -1
(-17x/9) < -1
-17x < -9
x > 9/17

Question 25

Algebra Strategy Guide, Ch 2, Q 8. a + b = 10, b + c = 12 and a + c = 16

Answer 25

2a + 2b + 2c = 38

a + b + c = 19
-(a + b + 0 = 10)
c = 9

b + 9 = 12
b = 3

a + 9 = 16
a = 7

Question 26

Algebra Strategy Guide, Ch 7, Q 10. If 0 < ab < ac, is a negative?

(1) c < 0
(2) b > c

Answer 26

Answer D. By the transitive property of inequalities, if 0 < ab < ac, then 0 < ac. Therefore, a and c must have the same sign.

(1) SUFFICIENT. c is negative. Therefore, a is negative
(2) SUFFICIENT. We are told that b > c and from the question stem, we know that ab < ac. Thus, when you multiply each side of b > c by a, the signs get flipped. For inequalities, the only circumstance that must be true in order to flip the sign when you multiply by something is if you are multiplying by a negative. Thus, we know that a must be negative.

Question 27

What is the rule for taking the reciprocal of inequalities?

Answer 27

*If you know the signs of the variables, you should flip the inequality UNLESS x and y have different signs

If x < y:

• 1/x > 1/y when x and y are positive. Flip the inequality
• 1/x > 1/y when x and y are negative. Flip the inequality
• 1/x < 1/y when x is negative and y is positive. Do NOT flip the inequality
• if you do not know the signs of x or y, you cannot take the reciprocals

Question 28

Given that ab < 0 and a > b, which of the following must be true?
I. a > 0
II. b > 0
III. 1/a > 1/b

Answer 28

I and III only. The question stem tells you that a and b have different signs (because ab < 0), so a must be positive and b must be negative (because a > b). Therefore, I is true while II is not true.

You also know from the reciprocal rule that if a > b, then 1/a < 1/b (the inequality sign is flipped) unless a and b have different signs, in which case 1/a > 1/b. Since a and b have different signs here, statement III is true as well.

Question 29

What is the rule for squaring inequalities?

Answer 29

(1) If both sides are known to be negative, then flip the inequality sign when you square
(2) If both sides are known to be positive, then you do not flip the inequality sign when you square
(3) If one side is positive and one side is negative, then you cannot square
(4) If the signs are unclear, then you cannot square

*As with reciprocals, you cannot square both sides of an inequality unless you know the signs of both sides of the inequality

Question 30

Is x^2 > y^2?

(1) x > y
(2) x > 0

Answer 30

Answer E.

(1) INSUFFICIENT. You do not know whether x and y are positive or negative numbers. For example, if x = 5 and y = 4, then x^2 > y^2. However, if x = -4 and y = -5 then x > y but x^2 < y^2
(2) INSUFFICIENT. Does not tell you anything about y.
(C) INSUFFICIENT. You know that x is positive and larger than y. This is still insufficient because y could be a negative number of larger magnitude than x. For example, if x = 3 and y = 2, then x^2 > y^2, but if x = 3 and y = -4, then x^2 < y^2

Question 31

Algebra Guide, Ch 11, Q 2. Is mn > -12?

(1) m > -3
(2) n > -4

Answer 31

Answer E. Inequalities.
(1) INSUFFICIENT. We do not know anything about n.
(2) INSUFFICIENT. We do not know anything about m.
(C) INSUFFICIENT. Combining the two statements, it is tempting to conclude that mn must either be positive or a negative number larger than -12. However, because either variable could be positive or negative, it is possible to end up with a negative number less than -12. For example, m could equal -1 and n could equal 50. In that case, mn = -50, which is less than -12.

Question 32

Algebra Guide, Ch 11, Q 5. Is x < y?

(1) 1/x < 1/y
(2) x/y < 0

Answer 32

Answer C. Inequalities.
(1) INSUFFICIENT. The meaning of this statement depends on the signs of x and y. If x and y are either both positive or both negative, then you can take reciprocals of both sides, yielding x > y. However, this statement could also be true if x is negative and y is positive. In that case, x < y
(2) INSUFFICIENT. The quotient of x and y is negative. In that case, x and y have different signs: one is positive and one is negative. However, this does not tell you which one is positive and which one is negative.
(C) SUFFICIENT. Combining the two statements, if you know the reciprocal of x is less than that of y, and that x and y have opposite signs, then x must be negative and y must be positive, so x < y

Question 33

Recognition: is x < z < y?

(1) x < 2z < y
(2) 2x < z < 2y

Answer 33

(1) Insufficient
(2) Insufficient
(C) Sufficient. Add inequalities when there are multiple variables and multiple inequalities facing the same direction

(x < 2z < y)
+(2x < z < 2y)
=3x < 3z < 3y
= x < z < y