Graphing linear inequalities on a number line might feel tricky at first, but don't worry! Once you learn the steps, it will be super easy. Let’s break it down so it’s simple to understand.

Step 1: Know the Inequality

Before you start graphing, you need to know what the inequality symbol means. Here are the main ones:

• $<$ means "less than"
• $>$ means "greater than"
• $\leq$ means "less than or equal to"
• $\geq$ means "greater than or equal to"

These symbols will change how your number line looks later.

Step 2: Find the Boundary Point

First, change the inequality into an equation to figure out your boundary point.

For example, if your inequality is $x < 3$, you’ll find the boundary at $x = 3$.

Step 3: Select the Right Circle

Now that you know your boundary point:

• If you see $<$ or $>$, use an open circle. This means the boundary point is not included in your answer.
• If you see $\leq$ or $\geq$, use a closed circle. This means the boundary point is included.

So, for $x < 3$, you draw an open circle on 3. For $x \leq 3$, use a closed circle.

Step 4: Shade the Right Area

Now here comes the fun part—shading! You need to figure out which way to shade.

• If your inequality is $x < 3$ or $x \leq 3$, shade to the left of 3. This shows all numbers less than 3.
• If you have $x > 3$ or $x \geq 3$, shade to the right instead.

Step 5: Check Your Work

Lastly, do a quick review. Make sure your circle is correct and you're shading the right way. Sometimes, it helps to pick a number from your shaded area and plug it back into the original inequality to see if it works.

Example

Let’s practice with an example. Suppose you want to graph the inequality $2x + 1 < 5$.

1. Solve for x:

   • Start with $2x + 1 < 5$.
   • Subtract 1 from both sides: $2x < 4$
   • Divide by 2: $x < 2$

2. Find the boundary: The boundary point is $x = 2$.

3. Choose your circle: Since it’s $<$, use an open circle on 2.

4. Shade the area: Shade to the left of 2.

5. Final check: Pick a point, like 1, and substitute back:

   • $2(1) + 1 < 5$, which simplifies to $3 < 5$. That’s true, so everything is correct!

Graphing linear inequalities can actually be fun! Once you practice a few times, you’ll feel great about it. So relax, grab a number line, and start shading!