Solve One-Variable Inequalities and Graph Solution Sets

Algebra and Function Practice

This lesson teaches how to solve one-variable linear inequalities such as 3x - 5 < 10, represent the solution set by shading a number line, and express it in interval notation. You will use inverse operations, remember to reverse the inequality symbol when multiplying or dividing by a negative number, and distinguish open and closed endpoints. The convention for shading is explicit: an open circle marks an excluded endpoint, a closed circle marks an included endpoint, and shading extends in the direction of all solutions. Letter styles and handwriting vary, so the rubric accepts any clearly drawn circle and arrow that matches the stated convention. All numerical examples are hypothetical and chosen for easy checking.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Solving Linear Inequalities with Inverse Operations

A linear inequality compares two expressions with <, ≤, >, or ≥. Solve it much like an equation: undo addition or subtraction first, then undo multiplication or division. The solution set is all values of the variable that make the inequality true.

The key difference from equations is the reversal rule: when you multiply or divide both sides by a negative number, reverse the inequality symbol. For example, -2x > 6 becomes x < -3 after dividing by -2 and flipping > to <.

1. Simplify each side if needed.
2. Add or subtract the same number from both sides to isolate the variable term.
3. Multiply or divide both sides by the coefficient; if it is negative, reverse the inequality symbol.
4. Check by substituting one value from the solution set and one value outside it.

Shading a Number Line with Open and Closed Endpoints

Use a number line to show the solution set visually. Place an open circle at an endpoint that is not included (< or >) and a closed circle at an endpoint that is included (≤ or ≥). Shade or draw an arrow from the endpoint in the direction of the solutions: right for greater than, left for less than.

For example, x ≥ 2 has a closed circle at 2 and shading to the right. The convention is stated so that any valid alternative, such as a thick arrow or a shaded segment, is acceptable if the endpoint type and direction are clear.

1. Draw a horizontal number line and mark the endpoint value.
2. Decide whether the endpoint is open or closed from the inequality symbol.
3. Shade or arrow to the right for > or ≥, and to the left for < or ≤.
4. Label the endpoint value clearly.

Writing Interval Notation

Interval notation uses parentheses for excluded endpoints and brackets for included endpoints. For x > 3, write (3, ∞). For x ≤ -1, write (-∞, -1]. Infinity always takes a parenthesis because it is not a number.

Match the number line to the notation: an open circle corresponds to a parenthesis, a closed circle to a bracket. Always write the smaller value first when both endpoints are finite.

1. Identify the endpoint and whether it is included.
2. Use a bracket for included endpoints and a parenthesis for excluded endpoints.
3. Use ∞ or -∞ with a parenthesis on the unbounded side.
4. Check that the direction of the interval matches the shaded direction.

Checking Solutions and Avoiding Common Errors

A quick check builds confidence. Pick a number inside the shaded region and substitute it into the original inequality; it should make a true statement. Pick a number outside the region; it should make a false statement. This catches reversal mistakes and endpoint errors.

Keep units consistent if the variable represents a measured quantity. All numerical values in this lesson are hypothetical examples chosen for simple arithmetic, not measurements of any specific equipment.

1. Substitute an inside value and confirm the inequality is true.
2. Substitute an outside value and confirm it is false.
3. Verify the endpoint type matches the symbol.
4. Verify the interval notation matches the number line.

Worked example

Solve 4x - 7 ≤ 9, graph the solution on a number line, and write it in interval notation.

Four practice exercises

Common mistakes

• Forgetting to reverse the inequality symbol when multiplying or dividing by a negative number. Circle the negative coefficient before dividing, then flip the symbol immediately and check with a test value.
• Using a closed circle for < or >, or an open circle for ≤ or ≥. Match the circle to the symbol: open for strict inequalities, closed for inclusive ones.
• Writing interval notation with brackets around infinity. Always use parentheses with ∞ and -∞ because infinity is not a finite endpoint.
• Shading the wrong direction on the number line. Test a value on each side of the endpoint in the original inequality to confirm the correct direction.

Self-check rubric

• Correctly applies inverse operations to isolate the variable.
• Reverses the inequality symbol only when multiplying or dividing by a negative number.
• Uses an open circle for < or > and a closed circle for ≤ or ≥.
• Shades or arrows in the correct direction on the number line.
• Writes interval notation with correct parentheses or brackets, including parentheses for infinity.
• Accepts valid alternative handwriting or shading styles if the endpoint type and direction are clear.

Questions and answers