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Graphing Inequalities

Q: Why do we use a dashed line for some inequalities?

A dashed line is used for strict inequalities () to show that the points on the boundary line itself are not included in the solution set. A solid line is used when the points are included ($\le$ and $\ge$).

Q: How do I graph an inequality like x < 5 on a coordinate plane?

This inequality has a vertical boundary line at $x = 5$. Since the symbol is $<$, the line should be dashed. You then shade the region where the x-values are less than 5, which is the entire area to the left of the line.

Q: What does the shaded region of the graph represent?

The shaded region represents the complete solution set of the inequality. Every single coordinate pair $(x, y)$ within the shaded area will make the original inequality a true statement.

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Graphing inequalities is a visual way to understand the infinite solutions that a problem can have. This guide will take you from simple inequalities on a number line to graphing two-variable linear inequalities on a coordinate plane, giving you the tools to master this essential algebra skill.

Graphing Inequalities — an original Algebra911 reference diagram defining graphing inequalities with its key formula and a worked example. — A Comprehensive Guide to Graphing Inequalities

What Are Inequalities and Their Graphs?

An inequality graph is a visual representation of all the possible solutions to an inequality. Unlike an equation, which typically has one or two solutions, an inequality describes a range of values. For example, the equation $x = 3$ has only one solution: $3$ . But the inequality $x > 3$ has infinite solutions: $4$ , $5.5$ , $100$ , and any other number greater than $3$ .

Because we can't list all the solutions, we use a graph to show them. This involves drawing a line and shading the part that contains all the solutions. There are four main inequality symbols you need to know:

$<$ : Less than
$>$ : Greater than
$\leq$ : Less than or equal to
$\geq$ : Greater than or equal to

The first two ( $<$ and $>$ ) are called strict inequalities because they do not include the boundary value. The second two ( $\leq$ and $\geq$ ) are inclusive inequalities because they do include the boundary value. This difference is crucial for how we draw our graphs.

How Do You Graph Inequalities on a Number Line?

For inequalities with only one variable (like $x$ or $y$ ), we use a simple number line to visualize the solutions. The process involves two key decisions: choosing the right kind of circle (open or closed) and shading in the correct direction.

1. The Circle: Open or Closed?

An open circle (an unfilled circle, like a donut) is used for strict inequalities ( $<$ and $>$ ). It shows that the number itself is not part of the solution.
A closed circle (a filled-in dot) is used for inclusive inequalities ( $\leq$ and $\geq$ ). It shows that the number is part of the solution.

2. The Shading: Left or Right?

For "less than" symbols ( $<$ or $\leq$ ), you shade to the left of the circle, where the numbers are smaller.
For "greater than" symbols ( $>$ or $\geq$ ), you shade to the right of the circle, where the numbers are larger.

Here is a handy table to summarize:

Symbol	Meaning	Circle Type	Shading Direction
$x < a$	x is less than a	Open	Left
$x > a$	x is greater than a	Open	Right
$x \leq a$	x is less than or equal to a	Closed	Left
$x \geq a$	x is greater than or equal to a	Closed	Right

Example 1

Graph the inequality $x \geq - 2$ on a number line.

Solution:

Identify the number: The number in our inequality is $- 2$ . We'll place our circle at this point on the number line.
Choose the circle type: The symbol is $\geq$ (greater than or equal to). The "or equal to" part tells us to use a closed circle.
Determine the shading direction: The symbol means "greater than," so we shade to the right of the circle, where all the numbers are larger than $- 2$ .
Draw the graph: Draw a number line, place a closed circle at $- 2$ , and draw a thick arrow extending to the right.

What Are Linear Inequalities in Two Variables?

A linear inequality in two variables is an inequality that involves both $x$ and $y$ , such as $y > 2 x + 1$ . Just like a linear equation (e.g., $y = 2 x + 1$ ) forms a straight line on a graph, a linear inequality also involves a line. However, this line acts as a boundary line.

The boundary line divides the coordinate plane into two distinct regions, or half-planes. The solution to the inequality isn't the line itself, but rather all the points on one side of the line. Our job when graphing is to find this boundary line and then determine which of the two regions represents the solutions.

The general forms of linear inequalities are:

Ax + By < C

Ax + By > C

Ax + By \le C

Ax + By \ge C

Often, it's easiest to work with these when they are in slope-intercept form, like $y > m x + b$ .

How Do You Graph a Linear Inequality on the Coordinate Plane?

Graphing a two-variable inequality involves a clear, four-step process. Let's walk through it.

Step 1: Graph the Boundary Line. Pretend the inequality symbol is an equals sign and graph the resulting linear equation. You can use the slope and y-intercept (if it's in $y = m x + b$ form) or find the x- and y-intercepts.
Step 2: Choose a Dashed or Solid Line. This step is crucial and depends on the inequality symbol.
- For strict inequalities ( $<$ or $>$ ), the points on the line are not solutions. We show this by drawing a dashed line.
- For inclusive inequalities ( $\leq$ or $\geq$ ), the points on the line are solutions. We show this by drawing a solid line.
Step 3: Pick a Test Point. Choose any coordinate pair $(x, y)$ that is not on the boundary line. The easiest point to use is almost always the origin, $(0, 0)$ , unless the line passes directly through it.
Step 4: Test and Shade. Substitute the coordinates of your test point into the original inequality.
- If the test point makes the inequality true, shade the entire region that contains the test point.
- If the test point makes the inequality false, shade the entire region on the opposite side of the line.

Example 2

Graph the inequality $y \leq - x + 3$ .

Solution:

Graph the boundary line: The equation is $y = - x + 3$ . The y-intercept is $3$ , and the slope is $- 1$ (down 1, right 1). Plot the y-intercept at $(0, 3)$ and use the slope to find another point, like $(1, 2)$ .
Choose the line type: The symbol is $\leq$ . The "or equal to" part means we need a solid line. Draw a solid line through the points.
Pick a test point: The line does not pass through the origin, so let's use $(0, 0)$ as our test point.
Test and shade: Substitute $x = 0$ and $y = 0$ into the original inequality: $y \leq - x + 3$ becomes $0 \leq - (0) + 3$ , which simplifies to $0 \leq 3$ . This statement is true. Since our test point worked, we shade the region that contains the origin $(0, 0)$ . This is the area below the boundary line.

Worked Example: When You Need to Rearrange First

Sometimes, the inequality isn't given in slope-intercept form. Your first task is to isolate $y$ on one side. Be very careful: if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality symbol.

Example 3

Graph the inequality $3 x - 2 y < 6$ .

Solution:

Isolate y: We need to get $y$ by itself.
$3 x - 2 y < 6$
Subtract $3 x$ from both sides:
$- 2 y < - 3 x + 6$
Now, divide both sides by $- 2$ . Since we are dividing by a negative number, we must flip the inequality symbol from $<$ to $>$ .
$y > \frac{- 3}{- 2} x + \frac{6}{- 2}$
$y > \frac{3}{2} x - 3$
Graph the boundary line: The equation is $y = \frac{3}{2} x - 3$ . The y-intercept is $- 3$ and the slope is $\frac{3}{2}$ (up 3, right 2).
Choose the line type: The symbol in our rearranged inequality is $>$ . This is a strict inequality, so we must use a dashed line.
Pick a test point: The origin, $(0, 0)$ , is not on the line, so it's a good choice.
Test and shade: Let's use the original inequality for our test: $3 x - 2 y < 6$ .
Substitute $x = 0$ and $y = 0$ :
$3 (0) - 2 (0) < 6$
$0 - 0 < 6$
$0 < 6$ This statement is true. Therefore, we shade the region that includes our test point, $(0, 0)$ . This is the area above the dashed line.

Key formulas for graphing inequalities by Algebra911.

What About Horizontal and Vertical Line Inequalities?

Inequalities with only one variable can also be graphed on the coordinate plane. They result in horizontal or vertical boundary lines.

Horizontal Lines: Inequalities like $y > c$ or $y \leq c$ (where $c$ is a constant) have a horizontal boundary line at $y = c$ . Shading is simple: for $>$ or $\geq$ , you shade above the line. For $<$ or $\leq$ , you shade below the line.
Vertical Lines: Inequalities like $x > c$ or $x \leq c$ have a vertical boundary line at $x = c$ . For $>$ or $\geq$ , you shade to the right of the line. For $<$ or $\leq$ , you shade to the left of the line.

For these simple cases, you don't necessarily need a test point, but you can still use one to confirm your shading is correct.

Example 4

Graph the inequality $x < 4$ on the coordinate plane.

Solution:

Graph the boundary line: The equation is $x = 4$ . This is a vertical line where every point has an x-coordinate of $4$ .
Choose the line type: The symbol is $<$ , so we use a dashed line.
Shade the correct region: Since it's "less than," we need to shade the side where the x-values are less than $4$ . This is the entire region to the left of the vertical line.

What Are Common Mistakes When Graphing Inequalities?

Graphing inequalities has a few tricky spots where students often make errors. Be on the lookout for these common mistakes:

Forgetting to Flip the Symbol: This is the most common mistake. When you multiply or divide both sides of an inequality by a negative number, you MUST reverse the inequality symbol (e.g., $>$ becomes $<$ ).
Mixing Up Dashed and Solid Lines: Remember the rule: solid lines for $\leq$ and $\geq$ (they include the "equal to" line), and dashed lines for $<$ and $>$ . A dashed line signals that the boundary is not part of the solution.
Shading the Wrong Side: Always use a test point like $(0, 0)$ to be certain. Don't just guess based on whether the symbol is "greater than" or "less than," as this can be misleading if you haven't isolated $y$ correctly.
Confusing Horizontal and Vertical Lines: An easy way to remember is that $x = c$ is a vertical line (it crosses the x-axis at $c$ ) and $y = c$ is a horizontal line (it crosses the y-axis at $c$ ).

Quick Summary: Your Graphing Checklist

When faced with a linear inequality, run through this checklist to make sure you get it right every time.

Isolate $y$ : If necessary, rearrange the inequality into slope-intercept form. Remember to flip the symbol if you multiply or divide by a negative!
Find the Boundary Line: Temporarily replace the inequality symbol with $=$ and graph this line.
Solid or Dashed?: Check the original symbol. Use a solid line for $\leq$ or $\geq$ . Use a dashed line for $<$ or $>$ .
Test a Point: Pick a point not on the line (usually $(0, 0)$ ). Substitute its coordinates into the original inequality.
Shade the Solution: If the test point is true, shade its region. If it's false, shade the other region.

Frequently Asked Questions

What's the main difference between graphing an equation and an inequality?

Graphing a linear equation results in a single line. Graphing a linear inequality results in a boundary line (which can be solid or dashed) and a shaded region that represents all possible solutions.

Why do we use a dashed line for some inequalities?

A dashed line is used for strict inequalities ( $<$ and $>$ ) to show that the points on the boundary line itself are not included in the solution set. A solid line is used when the points are included ( $\leq$ and $\geq$ ).

What is a 'test point' and why is it important?

A test point is any coordinate pair not on the boundary line that we use to determine which side to shade. By substituting it into the inequality, we can see if that region is part of the solution (true) or not (false), ensuring we shade correctly.

Can I use any point as a test point?

Yes, you can use any point as long as it is not on the boundary line. The origin $(0, 0)$ is the most popular choice because the math is easy, but if the line passes through the origin, you must pick a different point, like $(1, 1)$ or $(0, 1)$ .

What happens if I forget to flip the inequality sign when dividing by a negative?

If you forget to flip the inequality sign, your final inequality will be pointing in the wrong direction. This will cause you to shade the wrong side of the boundary line, leading to an incorrect graph.

How do I graph an inequality like x < 5 on a coordinate plane?

This inequality has a vertical boundary line at $x = 5$ . Since the symbol is $<$ , the line should be dashed. You then shade the region where the x-values are less than 5, which is the entire area to the left of the line.

What does the shaded region of the graph represent?

The shaded region represents the complete solution set of the inequality. Every single coordinate pair $(x, y)$ within the shaded area will make the original inequality a true statement.