Inequalities

Question 1

What does it mean to solve an inequality?

Answer 1

It means the same thing as to solve an equation: find the value or values of x that make the inequality true.
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.

Question 2

How to solve an inequality?

Answer 2

Isolate variable by transforming each side as you would normally do to an equation.

Question 3

How to pass multiplier / dividing terms to the other side of an inequality?

Answer 3

If you pass a positive multiplier / dividing term to the other side of an inequality, the sign of the inequality does not change.
x / 3 < 7
x < 7 * 3
x < 21

However, if you pass a negative multiplier / dividing term to the other side of an inequality, the sign of the inequality must be flipped
-2x > 10
x < 10 / -2
x < -5

Question 4

What is the absolute value of a number?

Answer 4

It describes how far that number is from 0. It is the distance between that number and 0 on a number line.
|5| = 5
|-5| = 5
Almost every absolute value is positive, except for 0.

Question 5

What to do when faced with a variable inside absolut value sign?
6 * |2x + 4| = 30

Answer 5

1) Isolate the absolute value on one side

|2x + 4| = 30 / 6
|2x + 4| = 5

2) Set up two equations (one positive and one negative)
+(2x + 4) = 5
2x = 1
x = 1 / 2

-(2x +4) = 5
-2x = 9
x = - 9 / 2

Question 6

What to do when faced with a variable inside absolut value sign in an inequality?

Answer 6

Drop the absolute value and set up two inequalities: one positive and one negative. Be attentive to the flip of the inequality sign when finding one of the possible values of the variable.
By the end, compare both solutions (draw arrows).
If the two solutions don’t overlap, write or between them and leave them as two inequalities. But if the solutions overlap, combine them into one big inequality.

|a| < 4
+(a) < 4
- (a) < 4
a < 4
a > -4
Overlap!
- 4 < a < 4