Solve And Graph Inequalities

🔑 Key Concepts

• Solving an inequality is similar to solving an equation, but the answer is a range of values instead of just one value.
• Use inverse operations to isolate the variable.
• If you multiply or divide by a negative number, you must reverse the inequality symbol.
• After solving, graph the solution on a number line using an open or closed circle.
• An open circle means the endpoint is not included, and a closed circle means the endpoint is included.

✏️ Worked Examples — Solving and Graphing Inequalities

Example 1

Step 1: Start with the inequality.

\(3x>-6\)

Step 2: Divide both sides by \(3\) to isolate \(x\).

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

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

Since we divided by a positive number, the inequality symbol stays the same.

Graph:
Use an open circle at \(-2\). Shade right because the solutions are greater than \(-2\).

Values that work:
\(-1,\ 0,\ 4\)

Values that do not work:
\(-2,\ -3,\ -10\)

Example 2

Step 1: Start with the inequality.

\(-10x\le40\)

Step 2: Divide both sides by \(-10\).

\(\displaystyle \frac{-10x}{\color{red}{-10}}\ge\frac{40}{\color{red}{-10}}\)

\(\boxed{x\ge-4}\)

Because we divided by a negative number, the inequality symbol reverses.

Graph:
Use a closed circle at \(-4\). Shade right because the solutions are greater than or equal to \(-4\).

Values that work:
\(-4,\ 0,\ 3\)

Values that do not work:
\(-5,\ -6,\ -10\)

Example 3

Step 1: Start with the inequality.

\(2x+4<-4\)

Step 2: Subtract \(4\) from both sides.

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

\(2x<-8\)

Step 3: Divide both sides by \(2\).

\(\displaystyle \frac{2x}{\color{red}{2}}<\frac{-8}{\color{red}{2}}\)

\(\boxed{x<-4}\)

Since we divided by a positive number, the inequality symbol stays the same.

Graph:
Use an open circle at \(-4\). Shade left because the solutions are less than \(-4\).

Values that work:
\(-5,\ -6,\ -10\)

Values that do not work:
\(-4,\ -3,\ 0\)

🧠 Math Vocabulary

• Inequality: a statement that compares two expressions using symbols such as \( < \), \( > \), \( \le \), or \( \ge \).
• Solution Set: all values that make the inequality true.
• Open Circle: shows that the endpoint is not included in the solution.
• Closed Circle: shows that the endpoint is included in the solution.
• Inverse Operations: operations used to undo addition, subtraction, multiplication, or division when solving.

💡 Main Idea

Solving inequalities follows many of the same steps as solving equations, but there is one major difference: when you multiply or divide by a negative number, the inequality symbol must reverse direction. After solving, the answer is graphed on a number line using an open circle if the endpoint is not included or a closed circle if it is included. The shading or arrow shows all values that make the inequality true.

📚 What You Should Already Know

Students should already know how to solve one-step equations, use inverse operations, and graph values on a number line. It also helps to understand the difference between greater than, less than, greater than or equal to, and less than or equal to.

🚀 What Comes Next

After solving and graphing one-step and two-step inequalities, students can move into multi-step inequalities, inequalities with variables on both sides, compound inequalities, and writing inequalities from real-world contexts.

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