3.2. Example

Today, we're going to discuss linear inequalities. Can anyone tell me how they differ from equations?

I think inequalities show a range of values rather than a single value.

Exactly! Inequalities use symbols like < or > instead of =. For example, instead of saying x = 3, we can say x < 5. This means x can be any value less than 5. To remember this, think of the 'inequality' as a 'range of possibilities.'

What about the other symbols like ≤ or ≥?

Great question! ≤ means 'less than or equal to', while ≥ means 'greater than or equal to'. So if we say x ≤ 3, x could be 3 or any number less than 3.

Let's summarize: Linear inequalities express ranges rather than exact values. Remember: < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to.

Now let's solve some linear inequalities in one variable. For example, how would we solve 2x - 3 < 5?

We can first add 3 to both sides.

Correct! What do we get after that?

2x < 8.

Yes! Now, how do we isolate x?

Divide both sides by 2 to get x < 4.

Excellent! Always remember, when you multiply or divide by a negative number, you must reverse the inequality. Let's summarize the key point: add or subtract first, then divide or multiply, and watch out for negatives!

Graphing is a powerful way to illustrate solutions of inequalities. If we have the inequality x < 4, how should we graph this?

We would put an open circle at 4 because it does not include 4.

Exactly! And which direction do we shade?

We shade to the left, because we're looking for any number less than 4.

Perfect! Now this visual representation is important, and you can use an open circle for < and > and a closed circle for ≤ and ≥. Let's recap: open circle for exclusive points and closed circle for included points. What did we learn?

How to graph inequalities using number lines!

Next, let's discuss systems of linear inequalities. If we have two inequalities, like x + y ≤ 6 and x ≥ 2, where do we graph this?

We graph each inequality separately and look for overlapping regions.

That's right! First, graph the boundary line from x + y = 6, using a solid line. Then for x ≥ 2, draw a solid vertical line at x = 2.

And we shade the region where both conditions are met.

Exactly! This overlapping area represents the solution to the system. So, we shade where both inequalities' solutions intersect. Key takeaway: always look for that overlap!