Inequalities - Humble Independent School District
Solving Inequalities – Chapter 3, Sections 4 and 5 (3.4, 3.5)
Solving Two-Step and Multi-Step Inequalities and Variables on Both Sides
We already saw that solving inequalities is just like solving equations with one-step (adding, subtracting, multiplying or dividing), so let’s look back at solving two-step . . .
Solving Equations by adding or subtracting and multiplying or dividing:
Here’s our ammunition:
1) we can add or subtract a number from both sides of an equation.
2) We can multiply or divide both sides of an equation by a number.
*and you always do them in this order!*
Now, we’ll need to do both.
Check it out:
Solve 2x – 7 = 3
Remember our goal: Get the x by himself!
There are two guys bugging the x... -7 and 2. The 2 is really locked on and the -7 is, kind of, hanging off… So, we’ll go after him first:
2x – 7 = 3
+7 +7
2x = 10
Now, ditch the 2 2x = 10
2 2
x = 5
Check it!
2x – 7 = 3… 2(5) – 7 = 3… 3=3
Yep!
Your turn:
Solve 3x + 8 = -7
Here’s one with a bit of a twist:
Solve 6 – x = 7
Ditch the 6: 6 – x = 7
-6 -6
-x = 1
x isn’t quite alone yet!
multiply both sides by -1 (-1)-x = 1 (-1)
x = -1
Solving Equations with Variables on BOTH SIDES:
Ok, so what if there are x’s on both sides?
9 – 3x = 5x + 6
Here’s the game: Get all x guys on one side and all number guys on the other.
*Ditch the smallest x guy first! (Trust me.)
Check it out:
9 – 3x = 5x + 6
+3x +3x .
9 = 8x + 6
-6 -6 ditch the 6
3 = 8x
3 = 8x ditch the 8
8 8
⅜ = x
Check it by sticking it back into both sides:
9 – 3x = 5x + 6
9 – 3(⅜) = 5(⅜) + 6
9 – 9/8 = 15/8 + 6
72 – 9 = 15 + 48
8 8 8 8
63/8 = 63/8 yep!
Your turn:
Solve 2x + 9 = 6x – 3
Solving Equations – Messier Ones
The game is the same… the problems just start out messier.
Let’s try one:
Solve 3(2x + 5) = 4x + 7 – x + 1
Start by cleaning up both sides…
3(2x) + 3(5) = 4x – x +7 + 1
Distribute combine like terms
6x + 15 = 3x + 8
-3x -3x ditch the smallest x
3x + 15 = 8
- 15 -15 ditch the 15
3x = 7
3x = 7
3 3 ditch the 3
x = -7/3
I’ll let you check it.
Here’s another one:
Solve 5x – 3 + 6x + 1 = 5 – 2(x + 4)
Clean up both sides:
11x – 2 = 5 -2(x) -2(4) combine like terms and distribute
11x – 2 = 5 – 2x – 8 multiply
11x – 2 = -2x – 3 combine 5 and -8
+2x +2x ditch the smallest x
13x – 2 = -3
+ 2 +2 ditch the -2
13x = -1
13 13 ditch the 13
x = -1/13
Your turn:
Solve 3 – (2x + 5) = 5 + x + 2(4x – 6)
Solving Inequalities – Messier Ones
It’s the exact same thing we’ve been doing, except our = sign looks like , ≤, ≥
All you need to do is remember the stuff we’ve already learned about inequalities:
Use the same steps as with solving equations, the only difference is the inequality symbol! It just stays the same, just like the equal sign doesn’t change, the inequality symbol doesn’t change. UNLESS:
*You multiply or divide an inequality by a negative number, *
then *FLIP THE SIGN!*
Assignment: pg. 191 # 1-15, pg. 197 # 1-19
Solving Inequalities – Special Cases
Vocabulary:
1) Identity – When solving an inequality, if you get a statement that is always true, the original inequality is an identity, and ALL REAL NUMBERS are solutions. (this means that you cancel out the variable and get an inequality that works, and infinitely many solutions [in this section, we say all real numbers instead] will be your answer)
2) Contradiction – when solving an inequality, if you get a false statement, the original inequality is a contradiction, and it has no solutions. (this means that you cancel out the variable and get an inequality that doesn’t work, so no solutions will be your answer)
Ok, so remember when we were solving equations and we got a problem where our variable cancelled out? Like this:
x + 2 + 5x = 6x – 3 + 7 – 2
6x + 2 = 6x + 2 combine like terms
-6x -6x move the smallest variable
2 = 2 hey, they cancelled out!
Now we ask, “is this a true statement?”
If it is – the answer is infinitely many solutions, if it’s not – no solution!
2 = 2 yep, that’s true!
So Infinitely many solutions is my answer!
With inequalities we do the same thing!
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