4.7 Graphs of Linear Inequalities

Graphing Linear Inequalities

Linear inequalities expand on linear equations, showing regions of solutions instead of single lines. They're useful for understanding real-world scenarios with ranges of acceptable values, like budgets or production limits.

Graphing these inequalities involves plotting a boundary line and shading the correct side. The inequality symbol determines which side to shade and whether the line is solid or dashed.

Graphing Linear Inequalities

How to graph a linear inequality

Here's the process, step by step:

1. Rewrite in slope-intercept form. Get the inequality into the form $y < mx + b$, $y > mx + b$, $y \leq mx + b$, or $y \geq mx + b$. This makes it easy to identify the slope and y-intercept for graphing.

2. Draw the boundary line using the slope ($m$) and y-intercept ($b$), just like you would for a regular linear equation.

   • Use a dashed line for strict inequalities ($<$ or $>$). The dashed line means points on the line are not included in the solution.
   • Use a solid line for inclusive inequalities ($\leq$ or $\geq$). The solid line means points on the line are included.

3. Shade the correct side of the boundary line:

   • Shade below the line for $<$ or $\leq$
   • Shade above the line for $>$ or $\geq$

"Above" and "below" refer to the standard orientation when the inequality is solved for $y$. If you're unsure, use a test point (covered below) to confirm.

Solution sets in inequality graphs

Unlike a linear equation where solutions lie on the line, a linear inequality's solutions fill an entire region. The shaded region represents the solution set, meaning every point $(x, y)$ in that region makes the inequality true.

• Any point in the shaded region satisfies the inequality when you substitute its coordinates for $x$ and $y$.
• Points on the boundary line are solutions only if the inequality is inclusive ($\leq$ or $\geq$).
  • For $y \leq 2x + 1$: the point $(1, 3)$ sits on the boundary line since $3 = 2(1) + 1$. Because the symbol is $\leq$, this point is a solution.
• Points on the boundary line are not solutions if the inequality is strict ($<$ or $>$).
  • For $y < 2x + 1$: that same point $(1, 3)$ is not a solution, because $3$ is not strictly less than $3$.

Testing points to verify your shading

If you're ever unsure which side to shade, the test point method gives you a definite answer.

1. Pick a test point that is clearly not on the boundary line. The origin $(0, 0)$ is the easiest choice because it simplifies the arithmetic. (If the boundary line passes through the origin, pick a different point like $(1, 0)$.)

2. Substitute the test point's coordinates into the original inequality.

3. Evaluate:

   • If the inequality comes out true, shade the side that contains the test point.
   • If the inequality comes out false, shade the opposite side.

For example, take $y > x - 2$. Plugging in $(0, 0)$: is $0 > 0 - 2$? That gives $0 > -2$, which is true. So you shade the side of the line that contains $(0, 0)$.

Now take $y < x - 2$. Plugging in $(0, 0)$: is $0 < 0 - 2$? That gives $0 < -2$, which is false. So you shade the side that does not contain $(0, 0)$.

Tools and concepts for graphing linear inequalities

• The coordinate plane is where you'll visualize the boundary line and shaded solution region.
• A graphing calculator can plot linear inequalities quickly, which is helpful for checking your work.
• You need a solid understanding of linear functions (slope, y-intercept, graphing lines) before tackling inequalities, since the boundary line is just the corresponding linear equation.