Write Inequalities from a Graph

✏️ Worked Example — Write an Inequality From Its Graph

1️⃣ Writing the Inequality

Question: Write an inequality that represents the shaded region on the graph.

We want the answer in slope-intercept form, so we are aiming for:

\(y > mx+b\) or \(y < mx+b\)

To do that, we need to answer these questions:

• What is the slope?
• What is the y-intercept?
• Is the region shaded above or below the line?
• Is the boundary line solid or dashed?

2️⃣ Read the Graph

• The line crosses the y-axis at \(1\), so \(b=1\).
• The line is dashed, so the inequality will be strict: either \(>\) or \(<\).
• The shaded region is above the line, so the inequality uses \(y >\).

So far we know the inequality has the form:

\(y > mx+1\)

3️⃣ Find the Slope and Write the Inequality

From the graph, start at the y-intercept \((0,1)\). The line goes up 2 and right 3.

\(m=\dfrac{\text{rise}}{\text{run}}=\dfrac{2}{3}\)

Substitute the slope and intercept into the inequality:

\(y > \dfrac{2}{3}x+1\)

Final Answer: Because the line is dashed and the region is shaded above it, the inequality is \(y > \dfrac{2}{3}x+1\).

🧠 Math Vocabulary

• Linear inequality: A statement that compares two expressions using symbols such as \(>\), \(<\), \(\ge\), or \(\le\).
• Boundary line: The line that separates the solution region from the non-solution region on the graph.
• Slope: The rate of change of a line, found by dividing rise by run.
• Y-intercept: The point where the line crosses the y-axis.
• Solution region: The shaded part of the graph showing all points that make the inequality true.

📚 What You Should Already Know

Before this lesson, students should know how to identify the slope of a line from a graph, locate the y-intercept, and recognize slope-intercept form \(y=mx+b\). Students should also be comfortable reading points on the coordinate plane.

🚀 What Comes Next

After writing and graphing single-variable and two-variable inequalities, students can move into systems of inequalities, where they graph more than one inequality on the same coordinate plane and identify the overlapping solution region.

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