Chapter 3, Section A3

Math Topics – Solving and Graphing Inequalities

Inequalities

How would you solve the inequality, ? If you had no idea what to do, what could be a good problem solving strategy? Think of your list from section 2.3:

• create a table of the information, or a list
• create a similar, smaller problem to try first
• look for a pattern
• guess/estimate and check (also called trial and error)
• work backwards
• draw a picture or make a physical model
• describe and solve the problem algebraically

For this problem, all of these ideas would work! Let’s try guess and check with a table, and then see if we can work backwards, draw a picture, and solve algebraically. We can start the table with the easiest numbers, 0, 1, 2 and 3, and then see if we need to go bigger or smaller.

So it looks like this means that numbers less than 3 work. That means the solution would be,  x < 3.

If we would like to try working backwards and algebra, this would mean solving the inequality.

If we know that 2x + 5 < 11, we know that 2x  < 6.

We can do the algebra step of subtracting 5 from both sides, as if we had an equation, but we can also tell because if we make a table, it looks like all the same numbers work.

2x + 5 < 11

   -5        -5

2x  <  6

if we next divide by 3 on both sides, yes, we get the same solution as what we found by guess and check:

2x  <  6

/2    /2

x < 3

To show these solutions on a number line, we would shade all the numbers less than three, and put a hollow dot, indicating that we do not include -3.

Notice that we shade the number line in the same direction that the < symbols points toward. Also notice that x < 3 means that all the numbers less than three are shaded, including negative numbers, and decimals. That is why, if we want our set to only include positive whole numbers, we must add that x ∈ ℕ.

Thus far, it looks as if we can solve inequalities the same way that we solve equalities, by doing the opposite operation, usually starting with undoing any addition and subtraction, and then any multiplication or division, working backwards by going in the opposite of the order of operations. The next two examples test out whether this method will always work to get the correct solution.

Example 1 Solve the inequality and graph the solution on the number line. Check whether your solution is correct using the original inequality.

first we “undo” any addition or subtraction by doing the opposite operation to both sides of the inequality.

+ 4        +4   Since 4 is being subtracted, we add 4

/5   /5 Since 5 is multiplying x, we divide by 5.

We shade all the numbers from 4 on up, using a filled in dot to show that we include 4, because we have a greater than or equal to symbol.

Now let’s check to see if our answer is correct by trying numbers that are greater than or equal to 4.

If we try 6, does that work in the original equation? What about 10?

gives us 30 – 4 = 26, which is greater than 16, so it is true that it is greater than or equal to 16.

gives us 50 – 4 = 46, which is greater than 16, so it is true that it is greater than or equal to 16.

What about the number 4, does that work?

gives us 20 – 4 = 16, which is equal to 16, so it is true that it is greater than or equal to 16.

Example 2 Solve the inequality and graph the solution on the number line. Check whether your solution is correct using the original inequality.

Example 3 will be two three part inequalities, one w negative

Ex 4 will have x in two places on one side, x in two places on opposite sides