Inequalities &Expressions

This lesson we will cover Inequalities with numbers and, later, with expressions.

Quick Review of our symbols and what they mean,

If we have the symbol > or < between two numbers a & b, it means one is greater than the other.

a > b means that a is a larger value than b, i.e. 2 > 1, 1000 > 0, -1 > -2

a < b means that b is larger than a, i.e. 1 < 2, 0 < 1000, -2 < -1

You can think of the sign as a pac man mouth that wants to eat the bigger number.

Key for Symbols: ≥ or ≤ means greater than or equal to. For a ≥ b or b ≤ a, this means that a is either greater than b or the same value as b. It is saying that a has a value that is the same value or bigger. e.g.

7 ≥ -3 (7 is greater than or equal to -3)

or 2 ≥ 2 (2 is greater than or equal or two)

or -3 ≤ -1 are all true (-3 is less than or equal to -1)

Integer line from -4 to 4

Negative numbers can be confusing, (intuitively, -99 seems a bigger value than -3). You can think of it as, the further to the right on the number line, the greater the value. You can also think of it in terms of temperature. For example, -10 F is colder than -5 F, so, -10 < -5

Question 1

Draw a number line for each of the statements and then determine if the statement is true.

6 > 1

First start off by writing what the question is saying in words. 6 is greater than 1.

Now look at our number line; As we can see on the number line, 6 is to the right of 1, so 6 it is greater than 1,

Therefore, the statement is true

1 and 6 highlighted on a number line

8 < 4

Write what it is saying, 8 is less than 4

As we can see, 8 is to the right of 4, so 8 is greater than 4,

Therefore, the statement is false.

1 and 6 highlighted on a number line

-9 < 6

Question says, -9 is less than 6,

Remember not to be thrown off by the negative values. Write our number line; -9 is to the left of 6, and therefore -9 is less than 6,

So, the statement is true

-9 and 6 highlighted on a number line

5 < - 4

Question says, 5 is less than -4,

Write our number line; 5 is to the right of -4, so 5 is greater than -4, and 5 is not less than -4,

Therefore, the statement is false

-4 and 5 highlighted on a number line

0 < -8

It is saying, 0 is less than -8,

Write our number line,

0 is to the right of -8 and therefore is greater than -8, [an easy mistake to think that 0 is a less value than a negative number, so be careful]

Therefore, the statement is false

-8 and 0 highlighted on a number line

-1 > -5

The question says, -1 is greater than -5,

-1 is to the right of -5 and is therefore greater than -5, as the questions says -1 is greater than -5,

The statement is true

-5 and -1 highlighted on a number line

Question 2

Given the inequality statement 4(x - 3) ≥ 2x + 2, insert the given values of x and determine whether or not the x value is true for the statement.

Hint

You can still think of these types of questions as if they were equations with “=” signs, like,

2x +1 = 3x - 2

Say you were checking for x = 3 to see if the statement was true, when you sub in the value of x, it is true if both sides “=” each other,

So, for x = 3, you would have,

2(3) + 1 = 3(3) - 2

7 = 7

You would say this equation is true for x = 3 as 7 = 7,

Now for these questions, you are doing the same thing, except with” >” or “<” or “≤” or “≥”, you are looking to see if the x value makes these inequalities true.

x = 10,

First sub x in as you would with a normal equation, remember to include the appropriate inequality symbol, but do not be thrown off by it,

4((10) - 3) ≥ 2(10) + 2

4(7) ≥ 2(7) + 2

28 ≥ 14 + 2

28 ≥ 16,

28 is greater than or equal to 16, so the statement is true for the value of x = 10

x = -3

Subbing in x = -3

4((-3) - 3) ≥ 2(-3) + 2

4(-6) ≥ 2(-3) + 2

-24 ≥ -6 + 2

-24 ≥ -4,

Being aware of the negative values, -24 is further to the left on the number line than -4, and therefore -24 is less than -4,

Therefore, the statement is false for x = -3

x = 7

4(x - 3) ≥ 2x + 2

Subbing in x = 7,

4((7) - 3) ≥ 2(7) + 2

4(4) ≥ 2(7) + 2

16 ≥ 14 + 2

16 ≥ 16,

16 is equal to 16, and therefore fits the statement of “greater than or equal to”, so,

The statement is true for x = 7

Question 3

For each question, sub in the value for x, and determine if it is true for the statement given.

3x - 1 < x + 5, x = 1

Sub in our x = 1 value

3(1) - 1 < (1) + 5

3 - 1 < 1 + 5

2 < 6,

2 is less than 6, so the statement is true when x = 1

4(x + 2) > {3x + 4} / 2, x = - 4

Sub in x = -4,

4((-4) + 2) > (3(-4) + 4) / 2

Doing the brackets first,

4((-4) + 2) > ((-12) + 4) / 2

4(-2) > (-8) / 2

-8 > -4,

Remembering our number line, -8 is less than -4,

Therefore, the statement is false for x = -4

x2 - 6x + 2 ≥ 12 + 3x, x = 2

Sub in x = 2

(2) 2 - 6(2) + 2 ≥ 12 + 3(2)

Exponents first,

4 - 6(2) + 2 ≥ 12 + 3(2)

Multiply,

4 - 12 + 2 ≥ 12 + 6

-6 ≥ 18,

We know, that 18 is greater than -6, so,

The statement is false for x = -4

{x - 3} / 6 ≤ {-3(x + 2)} / 7, x = -9,

Sub in x = - 9

((-9) - 3) / 6 ≤ (-3((-9) + 2)) / 7

Brackets first,

((-9) - 3) / 6 ≤ (-3(-7)) / 7

(-12) / 6 ≤ (21) / 7

-2 ≤ 3,

We know that 3 is greater than -2, so it fits the statement “-2 ≤ 3”,

Therefore, the statement is true for x = -9