CHAPTER 4-1
Introduction to INEQUALITIES and graphing them:
Writing an inequality
Graphing an inequality: open dot is < or >
Closed dot mean “less than or EQUAL” or “ greater than or EQUAL”
(think of the = sign as a crayon that you can use to COLOR IN THE DOT!)
The closed dot INCLUDES the number as part of the solution set.
For example, x ≥ -5 is any number greater than -5 but it also includes -5.
What’s the difference of inequalities from equations? Inequalities have many answers (most of the time an infinite number!)
Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution
GRAPHING INEQUALITIES:
First, graphing an equation's solution is easy
1) Say you found out that y = 5, you would just put a dot on 5 on the number line
2) But now you have the y ≥ 5
You still put the dot but now also darken in an arrow going to the right
showing all those numbers are also solutions
3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!
TRANSLATING WORDS:
Some key words to know: so make FLASH CARDS or do whatever you do to MEMORIZE these words!!
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes,
but I'm sure hoping for even less!
CHAPTER 4-2 Solving Inequalities with adding and subtracting:
Simply use the Additive Inverse Property as if you were balancing an equation!
The only difference is that now you have more than one possible answer.
Example: 5y + 4 > 29
You would -4 from each side, then divide by 5 on each side and get:
y > 5
Your answer is infinite! Any real number bigger than 5 will work!
See Graphing above to review how to graph the solution!!
CHAPTER 4-3 Solving inequalities with multiplication or division:
Again, you will use your equation skills,
but this time with the Multiplicative Inverse Property.
ONE MAJOR DIFFERENCE FROM EQUATIONS!
When you multiply or divide by a NEGATIVE to BALANCE,
you must SWITCH the inequality SYMBOL!
(Does not apply to adding or subtracting negatives.)
EXAMPLE: -3y > 9
You need to divide both sides by NEGATIVE 3 so the symbol
will switch from > to < in the solution
y < -3 is the answer
If you want to understand why:
3 < 10
Now multiply both sides by -1 (multiplication prop of equality)
You get -3 < -10, but THAT'S NOT TRUE!!!
You have to SWITCH THE SYMBOL to make the answer true: -3 > -10
REMEMBER: when you MULTIPLY or DIVIDE by a NEGATIVE,
the symbol SWITCHES
Chapter 4-4; DOING 2 STEPS WITH INEQUALITY SIGNS -
Same as equations except make sure you switch the symbol
if you multiply or divide by a negative!
Always finish with the variable on the left.
Check with whatever solution is easiest in the solution set!
With two steps:
Before you start, you may want to clear fractions or decimals, but if you don't mind using them, just get started with the checklist below. If you want to clear them, you should do that right after you distribute (between steps 1 and 2 below)
1. Do distributive property first (if necessary)
2 Combine like terms on each side of the wall (equal sign)
3. Jump the variables to one side of the wall (get all the variables on one side of the equation) by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term)
4. Add or subtract
5. Multiply or divide
6. Make sure the variable is on the LEFT side when finished.
By putting the variable on the left side of the inequality, you will be able to graph the solutions much easier. The arrow you will draw will follow the same direction as the inequality sign.
