Graphing a Linear Inequality in Slope-Intercept Form

💡 Main Idea

To graph \(y\ge\dfrac{1}{2}x-2\), first graph the related boundary equation \(y=\dfrac{1}{2}x-2\). Begin at the \(y\)-intercept \((0,-2)\), then use the slope \(\dfrac{1}{2}\) to plot additional points such as \((2,-1)\) and \((4,0)\). Draw a solid line because equality is included. Finally, shade above the boundary because the inequality represents all ordered pairs whose \(y\)-values are greater than or equal to the values on the line. A test point such as \((0,0)\) can be substituted into the inequality to verify that the correct region has been shaded.

✏️ Worked Examples — Graphing \( y \ge \dfrac{1}{2}x - 2 \)

Step 1: Graph the Boundary Line

Start with the inequality:

\( y \ge \dfrac{1}{2}x - 2 \)

The boundary line is:

\( y = \dfrac{1}{2}x - 2 \)

The \( y \)-intercept is \( -2 \), so plot the point \( (0,-2) \).

The slope is \( \dfrac{1}{2} \), which means rise 1 and run 2.

From \( (0,-2) \), you can plot points like \( (2,-1) \) and \( (4,0) \), then draw the line through those points.

Step 2: Decide Solid or Dashed

The symbol in the inequality is:

\( \ge \)

Because the inequality says greater than or equal to, the boundary line must be solid.

A solid line means the points on the line itself are also solutions.

For example, points such as \( (0,-2) \), \( (2,-1) \), and \( (4,0) \) lie on the line, so they satisfy the inequality.

If the symbol had been \( > \) instead, the line would be dashed and points on the line would not be solutions.

Step 3: Shade the Solution Region

Since the inequality is

\( y \ge \dfrac{1}{2}x - 2 \)

shade above the solid line.

The shaded region represents all points that make the inequality true.

The unshaded region below the line represents points that are not solutions.

🧠 Math Vocabulary

• Linear Inequality: An inequality that compares \( y \) to a linear expression.
• Boundary Line: The line you graph first to separate solutions from non-solutions.
• Solid Line: A boundary line used when the inequality includes equality, such as \( \le \) or \( \ge \).
• Dashed Line: A boundary line used when the inequality does not include equality, such as \( < \) or \( > \).
• Solution Region: The shaded area showing all points that make the inequality true.
• Unshaded Region: The part of the graph containing points that do not satisfy the inequality.

📚 What You Should Already Know

Before graphing this inequality, you should know how to graph a line using slope-intercept form, identify the \( y \)-intercept, and use slope to plot additional points. You should also understand that inequalities compare values using symbols such as \( > \), \( < \), \( \ge \), and \( \le \).

🚀 What Comes Next

After graphing a single linear inequality, the next step is interpreting graphs of inequalities, writing an inequality from a shaded graph, and then graphing systems of inequalities where the solution is the overlapping region.

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