Solving Inequalities

In Lessons 4-2, 4-3 and 4-4 we have learned about graphing inequalities, solving when addition and subtraction are involved, solving when multiplication is involved and when there is a mix of operations on one or both sides of the inequality sign.

We always want to get our final answer simplified into the form of:

x > 3

When we are solving for a variable we work backwards through the order of operations:

Subtraction or Addition first

Division or Multiplication second

Clear any addition or subtraction first.

X + 4 < 12

subtract 4 from both sides

X + 4 - 4 < 12 - 4


X < 8


Clear any multiplication or division.


3x > 21


Multiply by the reciprocal

1/3 • 3x > 21 • 1/3


x > 7


For more complex problems such as

2(x - 3) < 4(x - 5)

Use the distributive property to clear the parenthesis first.

2x - 6 < 4x - 20

Then combine like terms by moving the terms with the variable to one side of the inequality and the terms without the variable to the other.

2x - 2x - 6 < 4x - 2x - 20

-6 < 2x - 20



-6 + 20 < 2x - 20 + 20

14 < 2x

Then multiply by the reciprocal.

1/2 • 14 < 2x • 1/2

7 < x or x > 7


Remember to always simplify each side of the inequality before you begin combining like terms.

Graphing Inequalities on a Number Line

When graphing inequalities on a number line there are conventions for the direction of the line and the end point of the line.

The end point is closed (filled in) when the symbols are

≤ less than or equal to
or
≥ greater than or equal

The end point is open when the symbols are

< less than
or
> greater than

The direction of the line on the graph is determined by whether the amount is greater than (going to the right on the number line) or less than (going to the left on the number line).

If the inequality is written with the variable on the left then the inequality symbol points in the direction of the line.

Semester Exam

Congratulations to everyone for studying very diligently for the semester exam. We had our usual assortment of grades: A's, B's C's and for the first time this year, grades lower than C.

I know some of you are disappointed with your results and I have options for you that may help. They will not help change your grade for this semester but they might help you do better in the second semester. Remember you need at 85% for the year to move on to geometry in grade 9.

This blog is one of the things I am going to try to see if it helps you improve not just your grade but your UNDERSTANDING of the algebra concepts. I am not interested in your grade - although you and your parents may be. I am interested in you understanding our algebra concepts SO WELL that you can use them to work any problem.

Toward that end, I will be blogging about our math lessons and send you a link so you can read and think about the concepts.

I hope this is one more thing that will make a difference in your understanding of algebra!