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Example 1

Simplify the inequality:

-2x + 3y > x + 9

Again, we’re going to simplify this the same way we would an equation.

Now, because it is an inequality, we really want to simplify the inequality until we get y by itself.

This will help us graph it if we need.

So, to get y by itself, first we need to…..

+ 2x + 2x

And we get:

3y > 3x + 9

Now, we need to:

__ ___ __

3 3 3

And finally, we are left with:

y > x + 3

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But, wait, this looks familiar….

Didn’t we do this before?

Well, yes we did, but as more of a mention, not really as a lesson.

We went over rearranging an inequality to get y by itself, but you were never really told why to do that.

Just that you needed to.

However, now that we are tossing in a sequence of inequalities, you can now see why we make sure to solve for y.

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Example 2: Sequence example

Imagine you have the following:

Graph the following sequence:

3x + 3y > 12x – 9

y < 2x + 2

So, what should you do first?

Well, I don’t know about you, but I have no idea what the first inequality’s graph looks like.

So, how about we simplify it to slope intercept form first, then we can figure that out?

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So, let’s simplify

Again, we were given:

3x + 3y > 12x – 9

So let’s simplify!

First, since we are solving for y, let’s subtract 3x from both sides:

– 3x –3x

And we are left with:

3y > 9x – 9

Now, we divide by 3 on both sides

__ __ __

3 3 3

And finally we are left with:

y > 3x -3

So, now that we simplified the inequality, we can actually graph the sequence and see what answers the two share.

So, with the simplified inequality, we have:

y > 3x -3

y < 2x + 2

Again, first step is to choose one.

Let’s pick the one we just simplified:

y > 3x - 3

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Graphing our simplified inequality

Now that we’ve simplified it, let’s graph it.

So, again, our inequality was:

y > 3x - 3

First thing we need to do is graph the line: y = 3x - 3

Next, we notice that the inequality is greater than, not greater than or equal to, so we need to make sure our line is dashed.

Now, we need to shade.

Since y is on the left side of the inequality, and it is greater than, then we need to shade up.

Alright! We are half way done!

(Yes, I know, ugh)

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Now we graph the other inequality

So, now we need to graph the other inequality.

And of course, we need to make sure we graph the second inequality on the same graph.

So, let’s do:

y < 2x + 2

First thing we need to do is graph the line: y = 2x + 2

Next, we notice that the inequality is less than, not less than or equal to, so we need to make sure our line is dashed.

Now, we need to shade.

Since y is on the left side of the inequality, and it is less than, then we need to shade down.

And that’s all! �Let’s do another just in case though

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Example 3

Imagine you have the following:

Graph the following sequence:

Again, it looks like we have a very ugly looking inequality.

And, again, it’s next to impossible to graph that.

So, how about we simplify it first, then graph it?

At least we’ll get the right answer (and an A on the test!)

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So, let’s simplify

Again, we were given:

So let’s simplify!

First, since we are solving for y, let’s add 9 from both sides:

+9 + 9

And we are left with:

Now, we add 2x on both sides

+2x + 2x

And we are left with:

So, now that we simplified the inequality, we can actually graph the sequence and see what answers the two share.

Now, we divide by -5 on both sides

__ ____ ___

-5 -5 -5

Now, let’s make sure to switch the sign since we divided by a negative.

And we are left with:

So, with the simplified inequality, we have:

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Now let’s graph

So, to recap, we have:

We know we need to first graph the equation: y = 7x – 6

Now, since it’s less than or equal too, we know we need to keep the line solid.

�

So, now we look at the inequality.

Since y is on the left, and it’s less than or equal to, we know we need to shade down.

Now we graph the other one

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Now we graph the other inequality

So, now we need to graph the other inequality.

Again, remember we need to graph both on the same graph

So, let’s do:

First thing we need to do is graph the line: y = 4x - 6

Next, we notice that the inequality is greater than or equal to, so we can keep our line solid.

Now, we need to shade.

Since y is on the left side of the inequality, and it is greater than or equal to, then we need to shade up.

Again, the answer we are looking for is the shared shaded region.

Those are the answers that these inequalities share.

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Okay, enough review

So, how do we do the word problems?

Well, I’m so glad you asked! We will get started with them in a minute.

However, before we dive in head first, let’s go over some terminology, and maybe some translations as to what they mean as far as the equation goes.

So…

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Word Problem Translator

Alright, so after looking at a LOT of different word problems, I came up with this translator to help you guys out, as well as a few rules and tricks.

So of course, we have the original definitions, so let’s start with them:

Greater than – means >. It comes in between two things, like: x is greater than y (x > y)

Less than – means <. It comes in between two things, like: x is less than y (x < y)

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Some tricks/suggestions

Again, I get it, word problems are tough.

However, here are some tricks to help you through.

Pay attention to mentioned numbers – Many times when you see a number, either spelled out (for example, two instead of 2) or the actual number itself (for example, 2 instead of two), there’s a good chance that it represents either a variable, or what the end of the inequality.

- For example: George wants to get baseball and Pokémon cards for his friends. Baseball cards cost $5 a pack, and Pokémon cards cost $7 a pack. He wants to spend no more than $70. Write out the inequality.

b = baseball cards

p = pokémon cards

2. Watch out for the terminology – If you see phrases like “at least”, “at most”, “no more than” this is a clue that there may be more than one inequality.

-For example: George wants to get baseball and Pokémon cards for his friends. Baseball cards cost $5 a pack, and Pokémon cards cost $7 a pack. He wants to buy at least 20 items altogether, but he doesn’t want to spend more than $70. Write out the inequality.

b = baseball cards

p = pokémon cards

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Example 1

Solve for the following word problem:

Ada is purchasing roses for her new flower garden. She needs at least 15 different plants. The sunrise roses cost $3 each and the red roses cost $5 each. She wants to spend no more than $100. How many of each type can she buy? Determine the possibilities.

Alright, let’s take this in turns.

First, we can see that there are two words that mean two inequalities

no more than

at least

Now, let’s look for numbers and variables.

We can see that we have sunrise roses (s) that cost $3�So�3s

And we can see that we have red roses (r) that cost $5

Now we make our inequalities!

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Example 2

Solve for the following word problem:

Allan is purchasing golf balls for his golf buddies. The Callaway pack costs $30 and the Sergio Garcia pack costs $50 each. He wants to spend no more than $300 on gold balls, and he wants to buy at least 10 different packs. How many of each type can he buy? Determine the possibilities.

Alright, let’s take this in turns.

First, we can see that there are two words that mean two inequalities

no more than

at least

Now, let’s look for numbers and variables.

We can see that we have a Callaway pack (c) that costs $30�So�30c

And we can see that we have Sergio Garcia pack (g) that costs $50

50g

Now we make our inequalities!

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Example 3

Solve for the following word problem:

Kevin is purchasing video games for his favorite nephew. He begins looking around and finds that Steam games cost $60 each and Nintendo Switch games cost $70 each. He wants to spend no more than $600 on his favorite nephew, and he wants to buy at least 7 games. How many of each type can he buy? Determine the possibilities.

Alright, let’s take this in turns.

First, we can see that there are two words that mean two inequalities

no more than

at least

Now, let’s look for numbers and variables.

We can see that we have Steam games (s) that cost $60 each�So�60s

And we can see that we have Nintendo Switch games (n) that cost $70

70n

Now we make our inequalities!