Ch. 5 Inequalities and Absolute Values

Question 1

In an inequality, what are the multiplication/ division rules?

Answer 1

Can’t divide by 0 (same as equation)
If you divide/ multiply by negative, must flip sign

Question 2

In quant comparison, if you don’t know the signs of the variables in an inequality, what can you do?

Answer 2

You can try seeing what the solution would be in two cases: one where variable is +ve, one where variable is -ve.

Question 3

Can you add/ subtract inequalities?

Answer 3

Yes, same as system of equations, but signs must be facing same way.

Question 4

What’s a method for visualizing inequalities?

Answer 4

Create a number line!

Question 5

|x| ≤ 3

Answer 5

x ≤ 3
x ≥ -3

When you remove the absolute value bars, keep the inequality as-is. Then flip the sign and make the non-abs value part of the inequality negative.
OR you can keep the sign and make the abs value part of the inequality negative:
-x ≤ 3
-x (-1) ≤ 3 (-1)
x ≥ -3

Question 6

-6 < x < 6

Answer 6

-6 < x < 6
|x|< 6
x^2 < 36

Question 7

If a ≤ x ≤ b and c ≤ y ≤ d, how do you find the minimum or maximum value of xy?

Answer 7

Evalute ac, ad, bc, and bd, then determine min/ max.

Question 8

What do absolute value bars represent?

Answer 8

The distance from 0

Question 9

How to solve equations with absolute value bars

Answer 9

Isolate the expression inside the absolute value bars, then solve the equation twice: once for +ve case, once for -ve case.
If there is a variable on both sides, check your answers by plugging back in.
Remember |x| can NEVER = negative. So if you plug back in and abs value = -ve number, you know that’s an extraneous (false) solution.

Question 10

If |x| = |y|, what is true about x & y’s +ve/-ve?

Answer 10

Either x = y or x = -y
They’re either equals or opposites

Question 11

T or F: | x – 7 | = | 7 – x | ?

Answer 11

Yes.
|x| = |-x|

Question 12

What are the conditions for | a + b | = |a| + |b| ?

Answer 12

1. a & b must have same sign, OR
2. a and/or b = 0

Question 13

What are the conditions for | a - b | = |a| - |b| ?

Answer 13

1. a & b must have same sign and |a| ≥ |b|, OR
2. b = 0

Question 14

What is| a - b | ? |a| - |b|

Answer 14

It is | a - b | ≥ |a| - |b|

Question 15

What is | a + b | ? |a| + |b|

Answer 15

It is | a + b | ≤ |a| + |b|

Question 16

What happens if the absolute value of an expression = -ve?
For example, |x|= -2

Answer 16

There are no solutions; there is no way an absolute value can = -ve number.

Question 17

If you see both sides of an absolute value equation have a variable, what do you do?

Answer 17

You MUST check your solutions by plugging them back in and solving. There might be an extraneous solution (a solution that appears to be a true solution, but is not)

Question 18

What do you know about x if x^2 < |x|?

Answer 18

Remember, when x^2 < x < √x then you know 0 < x < 1
If x is absolute value, then it’s,
0 < x < 1 AND
-1 < x < 0

Basically you know it’s either a +ve or -ve proper fraction.