Inequalities (Part 4)
Our seven steps to solving inequalities:
What don’t we know? We don’t know x.
Get x to one side.
Now get x alone:
a. Handle any numbers that are being added or subtracted to x.
b. Then handle any numbers that are being multiplied or divided with.
Once we have x alone to one side, think of three values that x could be worth. Jot them down.
OPTIONAL: If you are a visual person, draw a number line. (See post about number lines.)
Now go back to the original question asking you to compare Quantity A and Quantity B. Put your finger on the bigger one.
a. If the bigger one is Quantity A, the answer is A.
b. If the bigger one is Quantity B, the answer is B.
c. If they are equal, the answer is C.
d. If you don’t know from the information given, the answer is D. (D for don’t know!)
OPTIONAL: Check your work.
Let’s work through this list of steps for this question:
1. What don’t we know? We don’t know x. Check.
2. Get x to one side. Right now it’s on both sides of the inequality, there’s an 8x on one side and a 3x on the other side. I’ll take the smaller one (3x) and subtract it from both sides. (Why do I take the smaller one? Just so we don’t end up with negative numbers at the end)
Now x is only on one side. We are on a roll!
3. Now get x alone:
a. Handle any numbers that are being added or subtracted to x. Well, the 11 is being subtracted from 5x, so I will add 11 to both sides.
b. Then handle any numbers that are being multiplied or divided with x. Since there is a 5 being multiplied by the x, I’ll divide both sides by 5.
4. Once we have x alone to one side, think of three values that x could be worth. Jot them down.
Well, x is less than 3, so x could be worth 2, 0, or -4.
5. OPTIONAL: If you are a number line person, draw a number line. (See post about number lines.)
Oh what the heck! Let’s do the optional step and draw a number line.
Why is there a donut hole over the 3? Because x is less than 3. (Only if it was less than or equal to 3, shown as x < 3, would we have a big solid dot over the 3.)
6. Now go back to the original question asking you to compare Quantity A and Quantity B. Put your finger on the bigger one.
a. If the bigger one is Quantity A, the answer is A.
b. If the bigger one is Quantity B, the answer is B.
c. If they are equal, the answer is C.
d. If you don’t know from the information given, the answer is D. (D for don’t know!)
Looking at our number line, or our sample values, we see that the answer is A. Quantity A is greater.
7. OPTIONAL: Check your work.
Let’s check ourselves! We are claiming above that x will NOT be equal to 3. So to check, we’ll make x equal to 3 in the original inequality and ensure things go horribly wrong.
8x – 11 < 3x + 4
8(3) – 11 < 3(3) + 4
24 – 11 < 9 + 4
13 < 13
That is not true! So x can’t be equal to 3. We can be confident in our answer that A, Quantity A (3) is greater than Quantity B (x).
Great work, you are getting the hang of this! The next post we’ll tackle absolute values.