Graphing Inequalities in Two Variables

This tutorial explains how to solve inequalities involving two variables using graphical methods. An inequality in two variables usually involves variables \(x\) and \(y\). An ordered pair \((a,b)\) is a solution if the inequality evaluates to true when \(x = a\) and \(y = b\).

Since each ordered pair corresponds to a point in the coordinate plane, the solution set of an inequality is generally a shaded region in the \(xy\)-plane.

The Graphical Method

  1. Graph the boundary: Replace the inequality symbol with an equal sign to find the corresponding equation. Graph this equation on the coordinate plane.
  2. Line style: Use a solid line if the inequality includes equality (\(\ge\) or \(\le\)), and a dashed line if it is a strict inequality (\(>\) or \(<\)).
  3. Test a point: Select a test point \((x, y)\) from one of the regions created by the boundary (the origin \((0,0)\) is often easiest if it doesn't lie on the boundary).
  4. Shade the solution: If the test point satisfies the original inequality, shade the region containing that point. If it does not, shade the opposite region.
Table of Contents

Examples with Graphical Solutions

Example 1: Solving a Vertical Linear Inequality

Show Solution

Problem: Solve graphically the inequality

\[ x \ge 2 \]

Solution

  1. Graph the corresponding equation \(x = 2\). This vertical line divides the plane into a left region and a right region.
  2. Because the inequality is \(\ge\) (greater than or equal to), the line \(x = 2\) is drawn as a solid line.
  3. Select a test point from one of the regions. Let's choose \((3,2)\), which lies to the right of the line.
  4. Substitute \(x = 3\) into the inequality: \[ 3 \ge 2 \]

This is a true statement. Therefore, the solution set is the entire region to the right of the line \(x = 2\), including the line itself.

Graphical solution showing a shaded region to the right of the solid vertical line x = 2

Example 2: Horizontal Boundary Line

Show Solution

Problem: Solve graphically the inequality

\[ y < 1 \]

Solution

  1. Graph the corresponding equation \(y = 1\), which is a horizontal line.
  2. Because the inequality is \(<\) (strictly less than), the boundary line is drawn as a dashed line.
  3. Select a test point below the line, such as \((1,-1)\).
  4. Substitute \(y = -1\) into the inequality: \[ -1 < 1 \]

This statement is true, meaning the solution set is the region below the horizontal line \(y = 1\). Points directly on the dashed line are not included.

Graphical solution showing a shaded region below the dashed horizontal line y = 1

Example 3: Standard Linear Inequality in Two Variables

Show Solution

Problem: Solve graphically the inequality

\[ 2y < -4x + 4 \]

Solution

  1. Graph the boundary line equation \(2y = -4x + 4\). Dividing by \(2\) gives \(y = -2x + 2\).
  2. The line has an \(x\)-intercept at \((1,0)\) and a \(y\)-intercept at \((0,2)\). Draw it as a dashed line because the inequality is strict (\(<\)).
  3. Select a test point that does not lie on the line, such as the origin \((0,0)\).
  4. Substitute \((0,0)\) into the original inequality: \[ 2(0) < -4(0) + 4 \] \[ 0 < 4 \]

This statement is true. The inequality is satisfied, so the solution set is the entire region containing the test point \((0,0)\) (the region below the line).

Graphical solution showing a shaded region below the dashed line 2y = -4x + 4

Verification: Test a point in the shaded region, e.g., \((0,-1)\): \(2(-1) = -2 < -4(0) + 4 = 4\) ✓

Example 4: Quadratic Inequality (Parabola Boundary)

Show Solution

Problem: Solve graphically the inequality

\[ y > x^2 - 2x - 3 \]

Solution

  1. Graph the corresponding equation \(y = x^2 - 2x - 3\). This forms an upward-opening parabola.
  2. By factoring \(y = (x-3)(x+1)\), we find the \(x\)-intercepts at \((3,0)\) and \((-1,0)\). The \(y\)-intercept is at \((0,-3)\). The vertex is at \((1, -4)\).
  3. Because the inequality is strict (\(>\)), draw the parabola as a dashed curve.
  4. Select a test point inside the curve, such as \((0,0)\).
  5. Substitute \((0,0)\) into the inequality: \[ 0 > (0)^2 - 2(0) - 3 \] \[ 0 > -3 \]

This statement is true. The solution set is the region containing \((0,0)\), which corresponds to the area above (inside) the upward-opening parabola.

Graphical solution showing the shaded region above the dashed parabola y = x^2 - 2x - 3

Verification: Test a point in the shaded region, e.g., \((1,0)\): \(0 > 1 - 2 - 3 = -4\) ✓

Frequently Asked Questions

How do you graph an inequality in two variables?

To graph an inequality in two variables: (1) Replace the inequality sign with an equal sign and graph the boundary line or curve. (2) Use a solid boundary for \( \le \) or \( \ge \) and a dashed boundary for \( < \) or \( > \). (3) Test a point not on the boundary. (4) Shade the region that contains the test point if it satisfies the inequality, otherwise shade the opposite side.

When do you use a solid line versus a dashed line?

Use a solid line (or curve) when the inequality includes equality (\( \le \) or \( \ge \)), meaning points on the boundary are part of the solution. Use a dashed line (or curve) for strict inequalities (\( < \) or \( > \)), meaning points on the boundary are excluded from the solution.

What is the test point method?

The test point method determines which region to shade. Choose any point not on the boundary line (often the origin \((0,0)\)). Substitute its coordinates into the original inequality. If the inequality is true, shade the region containing that point. If false, shade the opposite region.

How do you solve a quadratic inequality graphically?

Graph the corresponding parabola as the boundary (dashed for strict inequalities, solid for non-strict). Find the \(x\)-intercepts and \(y\)-intercept to sketch the parabola. Use a test point to determine which side (inside or outside the parabola) satisfies the inequality. For \( y > f(x) \), shade above the parabola; for \( y < f(x) \), shade below.

Is the origin always the best test point?

The origin \((0,0)\) is usually the easiest test point, but it cannot be used if it lies exactly on the boundary line or curve. In that case, choose any other convenient point such as \((1,0)\) or \((0,1)\).

References and Links on Inequalities