Graphing Inequalities

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Concept Review

Graphing Inequalities: Drawing Mathematical Boundaries

Imagine you're planning a movie night, but you need at least $20 after buying snacks to afford tickets. If you start with $35 and spend money on snacks, how much can you spend and still go to the movies? This is where inequalities become your best friend—they don't just give you one answer, they show you all your possibilities.

Unlike equations that have one specific solution, inequalities represent entire ranges of values that work. When we graph these on a number line, we're essentially drawing a map of all the mathematical solutions that satisfy our conditions.

Two-Step Inequalities: The Process

Let's solve that movie night problem mathematically. If you start with $35, spend s dollars on snacks, and need at least $20 left, we write: 35 - s ≥ 20

Step-by-Step Solution:

35 - s ≥ 20

35 - s - 35 ≥ 20 - 35 (subtract 35 from both sides)

-s ≥ -15

s ≤ 15 (divide by -1, flip the inequality sign!)

You can spend letter: 'W', title: 'Graphing Inequalities', concept: 5 or less on snacks. But how do we show all these possibilities visually?

Graphing on the Number Line

Think of a number line as a highway with mile markers. For s ≤ 15:

Draw a closed circle at 15 (because 15 is included in "≤")
Shade everything to the left (all values less than 15)
The arrow points left (toward negative infinity)

🔑 The Sign-Flip Rule

Here's what catches everyone off guard: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Think of it like directions on a map. If you're facing north and someone says "turn around," suddenly left becomes right and right becomes left. Multiplying by a negative flips the entire number line's perspective.

Open vs. Closed Circles

The type of circle you draw is like a velvet rope at a concert venue. A closed circle (●) means "this number is invited to the party"—use it for ≤ or ≥. An open circle (○) means "this number can look but can't touch"—use it for < or >.

🔑 Key Takeaway

Just like planning your movie night budget gave you a range of spending options, graphing inequalities shows you that math problems often have infinite solutions within specific boundaries. The number line becomes your visual guide to understanding not just one answer, but all possible answers that work.

Sample questions

1. Which number line graph represents x > 3?

○ Closed circle at 3, shaded to the right

○ Open circle at 3, shaded to the left

○ Closed circle at 3, shaded to the left

✓ Open circle at 3, shaded to the right

Answer: Open circle at 3, shaded to the right — Greater than uses open circle (not including 3) and shading to the right.

2. Graph the solution to 2x + 1 ≥ 7. What does the graph look like?

✓ Closed circle at 3, shaded to the right

○ Open circle at 3, shaded to the right

○ Closed circle at 3, shaded to the left

○ Open circle at 3, shaded to the left

Answer: Closed circle at 3, shaded to the right — Solve: 2x ≥ 6, x ≥ 3. Greater than or equal to uses closed circle at 3, shaded right.

3. The inequality x < -2 is graphed with:

○ Closed circle at -2, shaded left

○ Open circle at -2, shaded right

✓ Open circle at -2, shaded left

○ Closed circle at -2, shaded right

Answer: Open circle at -2, shaded left — Less than uses open circle, shading to the left (more negative).

Skills in this topic

Graph the solution set of a two-step inequality on a number line
Understand when to use an open circle vs. a closed circle
Write an inequality that represents a given graph
Graph compound inequalities on a number line
Identify equivalent inequalities based on their graphs

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