4.1. Example

Today, we will dive into linear inequalities, which help us represent relationships that are not fixed. Let's first clarify what a linear inequality is. Can anyone tell me the symbols we use?

Are they the less than and greater than signs?

Exactly! We use symbols like '<', '>', '≤', and '≥'. Now, who can explain how a linear inequality differs from a linear equation?

Is it because an equation shows equality while an inequality shows a range?

Great job! Remember, an equation tells us the exact solution while an inequality shows us options. A way to remember this is with the acronym 'SOLVE' - Solutions OR Limits, Values, Expressions. Let’s move to solving inequalities.

Now, let's look at how to solve a linear inequality in one variable. For instance, if we solve 2x - 3 < 5, what steps would we take?

First, we would add 3 to both sides?

Correct! Now, what do we get after that?

We get 2x < 8.

Right! And now, what’s the next step?

We divide by 2, so x < 4.

Nicely done! It's crucial to remember: when multiplying or dividing by a negative, we flip the inequality sign, which can be tricky. Let's summarize this: you can add, subtract, multiply, or divide both sides by a positive number without changing the inequality.

Next, we'll graph x < 4. Who remembers how we display this on a number line?

We use an open circle for <!

Correct! And after placing the circle at 4, where do we shade?

We shade to the left because x is less than 4.

Excellent! To remember, think of the phrase 'Less is Left'. This will help ensure you shade in the correct direction. Now let’s see how this works for inequalities in two variables.

Consider the inequality x + y ≤ 5. What’s our first step to graph it?

We rewrite it to find y?

Yes! We can write it as y ≤ -x + 5. Who can tell me how we’ll draw the boundary line?

Since it's ≤, we use a solid line?

Exactly! Next, we choose a test point. Let’s use (0, 0). What happens when we plug it in?

That gives 0 ≤ 5, which is true, so we shade that region!

Well done! Remember, the area you shade represents all possible solutions to the inequality.

Lastly, let's discuss a word problem: A student has $20 to spend on snacks, with each snack costing$2. How would you express this as an inequality?

We can let x be the number of snacks. So it's 2x ≤ 20.

Correct! And what does that simplify to?

Dividing by 2 gives x ≤ 10, meaning the student can buy up to 10 snacks.

Great job everyone! Remember, inequalities help us model scenarios where we have limits and constraints. The key takeaway is to always translate real situations into mathematical expressions accurately.