Graph Inequalities on a Number Line

📝 Practice Worksheets

Solving Inequalities Practice Multi-Step Inequalities Solving and Graphing Inequalities

🛠️ Related Tool

Inequality Graphing Explorer

🔑 Key Concepts

Use an open circle for inequalities containing \(<\) or \(>\).
Use a closed circle for inequalities containing \(\le\) or \(\ge\).
Shade left for values less than the boundary and right for values greater than the boundary.
Solve inequalities using the same inverse operations used to solve equations.
Reverse the inequality symbol when multiplying or dividing both sides by a negative number.
Check that the symbolic inequality and number-line graph describe the same solution set.

✏️ Worked Examples — Solving and Graphing Inequalities

Example 1 — Graph \(x>13\)

Problem: Graph the inequality \(x>13\) on a number line.

The symbol \(>\) means that the solution contains all numbers greater than \(13\), but it does not include \(13\).

Open circle: The boundary value \(13\) is not included.

Arrow direction: The arrow points right because numbers greater than \(13\) are located to the right.

Example 2 — Solve \(-3x+2\ge-1\)

Problem: Solve the inequality \(-3x+2\ge-1\) to find all possible values of \(x\).

Subtract \(2\) from both sides:

\(-3x+2\ {\color{red}{-2}}\ge-1\ {\color{red}{-2}}\)

\(-3x\ge-3\)

Divide both sides by \(-3\). Because the divisor is negative, reverse the inequality symbol:

\(\displaystyle \frac{-3x}{\color{red}{-3}}\le\frac{-3}{\color{red}{-3}}\)

\(\boxed{x\le1}\)

The solution includes \(1\) and every number less than \(1\).

Example 3 — Solve and Graph \(-6x\le18\)

Problem: Solve \(-6x\le18\), and graph all possible solutions on a number line.

Divide both sides by \(-6\). Because the divisor is negative, reverse the inequality symbol:

\(\displaystyle \frac{-6x}{\color{red}{-6}}\ge\frac{18}{\color{red}{-6}}\)

\(\boxed{x\ge-3}\)

Use a closed circle at \(-3\) because the boundary value is included. Shade right because the solutions are greater than or equal to \(-3\).

Example 4 — Solve and Graph a Fraction Inequality

Problem: Solve the inequality and graph its solution:

\(\displaystyle -\frac75>x+\frac1{10}\)

Rewrite the inequality with the variable expression first:

\(\displaystyle x+\frac1{10}<-\frac75\)

Subtract \(\displaystyle \frac1{10}\) from both sides:

\(\displaystyle x+\frac1{10}\ {\color{red}{-\frac1{10}}}<-\frac75\ {\color{red}{-\frac1{10}}}\)

\(\displaystyle x<-\frac{14}{10}-\frac1{10}\)

\(\displaystyle \boxed{x<-\frac{15}{10}=-\frac32}\)

Use an open circle at \(-1.5\) and shade left because the solution contains values less than \(-1.5\).

Example 5 — Solve \(-4(h+1)>8\)

Problem: Solve the inequality \(-4(h+1)>8\) to find all possible values of \(h\).

Divide both sides by \(-4\). Reverse the inequality symbol because the divisor is negative:

\(\displaystyle \frac{-4(h+1)}{\color{red}{-4}}<\frac8{\color{red}{-4}}\)

\(h+1<-2\)

Subtract \(1\) from both sides:

\(h+1\ {\color{red}{-1}}<-2\ {\color{red}{-1}}\)

\(\boxed{h<-3}\)

The solution includes every value less than \(-3\). For example, \(h=-4\) makes the original inequality true.

Example 6 — Solve and Graph \(3x+2<-4\)

Problem: Solve the inequality \(3x+2<-4\), and graph the solution on a number line.

Subtract \(2\) from both sides:

\(3x+2\ {\color{red}{-2}}<-4\ {\color{red}{-2}}\)

\(3x<-6\)

Divide both sides by \(3\):

\(\displaystyle \frac{3x}{\color{red}{3}}<\frac{-6}{\color{red}{3}}\)

\(\boxed{x<-2}\)

Use an open circle at \(-2\) because \(-2\) is not included. Shade left because the solutions are less than \(-2\).

🧠 Math Vocabulary

Inequality: A mathematical statement comparing two expressions using \(<\), \(>\), \(\le\), or \(\ge\).
Solution set: All values that make an inequality true.
Boundary value: The number where the graph of an inequality begins.
Open circle: A number-line symbol showing that the boundary value is not included.
Closed circle: A number-line symbol showing that the boundary value is included.
Inverse operations: Operations that undo one another and help isolate a variable.
Equivalent inequalities: Inequalities that have the same solution set.
Reverse the inequality: Change the symbol’s direction when multiplying or dividing both sides by a negative number.

💡 Main Idea

Solving an inequality involves isolating the variable while keeping both sides balanced. Most steps are the same as solving an equation, but multiplying or dividing by a negative number reverses the inequality symbol. The final solution represents a set of possible values and can be shown on a number line with an open or closed circle and an arrow indicating the direction of the solutions.

📚 What You Should Already Know

Students should know how to solve one-step and multi-step equations, apply integer and fraction operations, use inverse operations, and compare numbers on a number line. They should also recognize the meanings of less than, greater than, less than or equal to, and greater than or equal to.

🚀 What Comes Next

Students can extend these skills to inequalities involving distribution, combining like terms, variables on both sides, and real-world situations. They will also write inequalities from verbal descriptions and interpret solution sets within the context of a problem.

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