3.1-3.2 Solving Inequalities Using Addition & Subtraction.

3.1-3.2 Solving Inequalities

Using Addition & Subtraction

An inequality is like an equation, but instead of an equal sign (=) it

has one of these signs:

< : less than≤ : less than or equal to

> : greater than≥ : greater than or equal to

“x < 5”

means that whatever value x has, it must be less than 5.

Try to name ten numbers that are less than 5!

Numbers less than 5 are to the left of 5 on the number line.

0 5 10 15-20 -15 -10 -5-25 20 25

• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.• There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. • The number 5 would not be a correct answer, though, because 5 is not less than 5.

The open dot means that 5 is not a

solution

The number line is shaded to the left of 5, so all numbers less than

5 are solutions

“x ≥ -2”means that whatever value x has, it must be greater than or

equal to -2.

Try to name ten numbers that are greater than or equal to -2!

Numbers greater than -2 are to the right of 5 on the number line.

0 5 10 15-20 -15 -10 -5-25 20 25

• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.• There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. • The number -2 would also be a correct answer, because of the phrase, “or equal to”.

-2

The closed dot means that -2 is a solution

The number line is shaded to the right of -2, so all numbers greater

than -2 are solutions

Where is -1.5 on the number line? Is it greater or less than -2?

0 5 10 15-20 -15 -10 -5-25 20 25

• -1.5 is between -1 and -2.• -1 is to the right of -2.• So -1.5 is also to the right of -2.

-2

Solve an Inequality

w + 5 < 8We will use the same steps that we did with equations, if a number is added to the variable, we add the opposite sign to both sides:

w + 5 + (-5) < 8 + (-5)

w + 0 < 3

w < 3

All numbers less than 3 are

solutions to this problem!

More Examples

8 + r ≥ -2

8 + r + (-8) ≥ -2 + (-8)

r + 0 ≥ -10

w ≥ -10

All numbers from -10 and up (including -10) make this problem true!

More Examples

x - 2 > -2

x + (-2) + (2) > -2 + (2)

x + 0 > 0

x > 0

All numbers greater than 0 make this problem true!

More Examples

4 + y ≤ 1

4 + y + (-4) ≤ 1 + (-4)

y + 0 ≤ -3

y ≤ -3

All numbers from -3 down (including -3) make this problem true!

3.3 Solving Inequalities byMultiplying or Dividing

Objective: Solve inequalities by using the multiplication and division

properties of inequalities.

5-Minute Check

Solve each inequality.1. r – 9 < 4

2. s + 24 > 23

3. t – (-30) 40

4. u + (-14) -29

5. 19 < v – 9 v > 28

3.3 Solving Inequalities byMultiplying or Dividing

When you multiply or divide each side of an inequality by a positive integer, the result remains true.

3.3 Solving Inequalities byMultiplying or Dividing

Example 1Solve 9x > -36 and graph the solution on a number line.

3-7 Solving Inequalities byMultiplying or Dividing

Example 1Solve 9x > -36 and graph the solution on a number line. 9x > -36

3-3 Solving Inequalities byMultiplying or Dividing

Example 1Solve 9x > -36 and graph the solution on a number line. 9x > -36 9x/9 > -36/9 Divide each side by 9.

3-3 Solving Inequalities byMultiplying or Dividing

Example 1Solve 9x > -36 and graph the solution on a number line. 9x > -36 9x/9 > -36/9 Divide each side by 9. x > -4

3-3 Solving Inequalities byMultiplying or Dividing

Example 1Solve 9x > -36 and graph the solution on a number line. 9x > -36 9x/9 > -36/9 Divide each side by 9. x > -4

Notice that -2 is

included

3-3 Solving Inequalities byMultiplying or Dividing

Example 2Solve -4x 12 and graph the solution on a number line.

3-3 Solving Inequalities byMultiplying or Dividing

Example 2Solve -4x 12 and graph the solution on a number line. -4x 12

3-3 Solving Inequalities byMultiplying or Dividing

Example 2Solve -4x 12 and graph the solution on a number line. -4x 12 -4x/(-4) 12/(-4) Divide each side by –4 and

reverse the order symbol.

3-3 Solving Inequalities byMultiplying or Dividing

Example 2Solve -4x 12 and graph the solution on a number line. -4x 12 -4x/(-4) 12/(-4) Divide each side by –4 and x -3 reverse the order symbol.

3-3 Solving Inequalities byMultiplying or Dividing

Example 2Solve -4x 12 and graph the solution on a number line. -4x 12 -4x/(-4) 12/(-4) Divide each side by –4 and x -3 reverse the order symbol.

Question?• How is the inequality symbol related to the

shading on the number line?

When the inequality is written with variable on the left, the inequality symbol points in the direction of the shading.

3-3 Solving Inequalities byMultiplying or Dividing

Assignment: do c, e, f in your class folder

You must solve, check your answer

3.4 solve and Graph Multi-Step Inequalities

Classwork

• Complete Solving Inequalities pages 1 -2 in your classwork folder.

• You must solve, check and graph