Inequalities

Question 1

Inequalities with 0

Answer 1

In inequalities with 0, consider using positive/negative analysis

Common inequality statements there:

xy > 0 - i.e. both x and y are either positive or negative
xy

Question 2

Multiplying or Dividing Inequalities by Variables

Answer 2

Don’t multiply or divide inequalities by variables unless you know the sign of the number the variable stands for because if you don’t know the sign you don’t know if the inequality needs to be flipped or not.

Question 3

Convert Multiple Inequalities to Compound Inequality:

Answer 3

Bring inequalities into similar form by simplifying and rearranging so that all inequality symbols show in same direction, then line up common variables.

E.g. x > 8, x

Question 4

0

Answer 4

ab is positive, so a/b must be positive since a and be have the same signs.
Only because ab

Question 5

Adding Inequalities:

Answer 5

In order to add inequalities, the inequality signs must face in same direction.

E.g. Is a + 2b b

Both statements allow are not sufficient because they each miss information about other variables.

When looking at them together first line the statements up so the inequality signs show to the same direction, then add them up until you can answer the question. Like here sometimes you have to add certain inequalities twice.

       a

Question 6

Square-Rooting Inequalities

Answer 6

Just like equations involving even exponents, inequality problems involving even exponents require you to consider two scenarios too.

E.g. x^2 - 2

REMEMBER: you can only take square root of inequality if both sides are NOT negative because you can’t take the square root of a negative number. So, you can only take square root if you know the square of a variable is not negative and the value on the other side is not negative either

Question 7

Optimization Problems:

Answer 7

In these problems you have to focus on the largest and smallest possible values for each of the variables and test them.

E.g. If 2y + 3 ≤ 11 and 1 ≤ x ≤ 5, what is the maximum possible value for xy?

1. Look at highest and lowest values of x and y to find solution.

Extreme values for x:

Lowest value: 1
Highest value: 5

Extreme values for y:

2y + 3 ≤ 11
y ≤ 4

Indefinite for lowest value
Highest value: 4

2. Now look where you get the maximum value for xy, combining any possible x with any possible y to find the highest value possible:

Using y’s lowest value: Since y has no lowest value and x is positive, xy has no lowest limit.

Using y’s highest value: 4 is y’s highest value, so if you use x’s lowest value you get for xy: 4 x 1 = 4. If you use x’s highest value, you get for xy: 4 x 5 = 20.

So xy is maximized when x = 5 and y = 4.

Another example:

If -7 ≤ a ≤ 6 and -7 ≤ b ≤ 8, what is the maximum possible value for ab?

Extreme values for a: - 7 and 6
Extreme values for b: - 7 and 8

Looking at all combinations for axb you can see that – 7 x – 7 will yeald the highest value, 49.
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Inequalities and Absolute Value:

One way to think about simple absolute value expressions in inequalities is the distance from 0. It helps to graph the solution.

E.g. in |x|

Question 8

Reciprocals and Inequalities:

Answer 8

If you don’t know the signs of the variables you can’t take reciprocals. If you know the signs, flip the inequality when taking the reciprocal unless you the variables have different signs.

E.g. if x 1/5

• If x and y are both negative then flip the inequality: e.g. – 5 - 1/3
• if x is negative and y is positive, don’t flip inequality because one expression will be negative and one positive even after taking reciprocal so inequality still is the same: e.g. – 3

Question 9

Squaring Inequalities:

Answer 9

You can’t square both sides of inequality unless you know both sides.

Rules:

• If both sides are known to be negative then flip the inequality when squaring: e.g. if x 9. But if x > - 3 then x could be negative or positive. If negative then you can flip the inequality because both sides are negative and so x^2 9 then x^2 > 9. But if x y^2.