Inequalities and Absolute Values

Question 1

Inequality Addition

Answer 1

You can only add inequalities when their signs are in the same direction:

• If a > b and c > d (signs in same direction: > and >) –> a+c > b+d
• Example: 3 < 4 and 2 < 5 –> 3+2 < 4+5

Question 2

Inequality Subtraction

Answer 2

You can only subtract inequalities when their signs are in the opposite directions:

• If a > b and c < d (signs in opposite directions: > and <) –> a-c > b-d
• Example: 5 > 1 and 3 < 4 → 5-3 > 1-4

Question 3

Inequality Multiplication

Answer 3

If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.

• x < a and y < b
• xy < ab

Question 4

Inequality Division

Answer 4

But if both sides of both the inequalities are positive and the signs of the inequality are opposite, then you can divide them

• x < a and y > b
• x/y < a/b → final sign takes the sign of the numerator

Question 5

Never multiply or divide an inequality by a variable unless the sign of the variable is known

Answer 5

px < 5x
→ p < 5 or
→ p > 5

Question 6

Compound Inequality

Answer 6

When a compound inequality is divided or multiplied by a negative number, both of the signs must be reversed.

Question 7

Multiple Inequalities

Answer 7

Set up a number line
e.g., If a > b, c > a, and d < b, what’s true?

Question 8

Simplifying Inequalities with x²

Answer 8

If x² > b and b is positive, then | x | > √b
→ x > √b or x < -√b

If x² < b and b is positive, then | x | < √b
→ -√b < x < √b

Question 9

The min or max value of xy

Answer 9

If a ≤ x ≤ b and c ≤ y ≤ d, to find the max and min value of xy, evaluate: ac, ad, bc, bd

• Max = the largest of these four
• Min = the smallest of these four

BE CAREFUL with the inequality signs. If a < x < b and c < y < d and x and y are integers, choose values that are within the range, not a, b, c, d!

Question 10

If 2 absolute values are equal, it must be true that the expressions with the absolute value bars are either equal or opposite.

Answer 10

1. 16x + 14 = 8x + 6
2. 16x + 14 = -(8x + 6) → No need to test the other way around

Data sufficiency question: Does | r + s | = | m + n |?

1. r + s = -m - n
2. r + s = m + n

Answer = D.

16x + 14 | = | 8x + 6 | → Test 2 scenarios

Question 11

Adding absolute values: | a + b | ≤ | a | + | b |

Answer 11

If | a + b | = | a | + | b |, either…

1. One or both quantities is 0; or
2. Both quantities are of the same sign

a + b | ≤ | a | + | b | is always true

Question 12

Subtracting absolute values: | a - b | ≥ | a | - | b |

Answer 12

If | a - b | = | a | - | b |

1. The 2nd quantity is 0; or
2. Both quantities are of the same sign and the absolute value of the 1st quantity is ≥ the absolute value of the 2nd quantity

→ If b ≠ 0 and | a - b | = | a | - | b |, then a and b must have the same sign and | a | ≥ | b |

a - b | ≥ | a | - | b | is always true
→ a and b can be either positive or negative

Question 13

If | x | < b

Answer 13

Then -b < x < b

Question 14

If | x | > b

Answer 14

Then x > b or x < -b

Question 15

If the absolute value of an expression is equal to a negative number…

Answer 15

There’s no solution

e.g., | x + 1 | = -2 → No need to solve.

BE CAREFUL: | x - 1 | = 2x → Solve it, but CHECK your answers

Question 16

If x² = | x |

Answer 16

x must be -1, 0, or 1

if x is none of these 3 values, then x² > | x | or x² < | x |

Question 17

Is x² > y²

Answer 17

Simplify and look at signs → (x+y)(x-y) > 0

Question 18

Inequality Operations (example 1)

Answer 18

If 5a > 2b, is the sum of a and b greater than 0?

1. 2b > 4a
2. b > 0

Answer: D. For #1:

• 5a - 2b > 0 and 2b - 4a > 0
• Add them up → a > 0
• 2b > positive #, so b > 0 as well

Question 19

Inequality Operations (example 2)

Answer 19

If x > y > 0, is x > 3?

1. 2/x + 1/y = 11/15
2. 2x + 3y = 16

Answer = D. Process = solve for y in terms of x, then substitute y in the x > y inequality