Writing and Graphing One-Variable Inequalities: Phrase to Number Line

Algebra and Function Practice

This lesson teaches how to turn a verbal phrase into a one-variable inequality, solve it using inverse operations, and represent the solution set on a number line. The key graphing convention is explicit: a closed circle marks a value included in the solution, an open circle marks a value excluded, and an arrow shows the direction of all remaining solutions. Because handwriting and letter styles vary, the rubric accepts any clearly drawn circle and arrow that follows this convention, not one universal shape. All numerical values used in examples are hypothetical and chosen for easy arithmetic; measure your own equipment if you adapt the tasks.

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

From Words to Symbols

A verbal phrase becomes an inequality by matching comparison words to symbols. Phrases such as at least, no less than, and minimum mean greater than or equal to (≥). Phrases such as at most, no more than, and maximum mean less than or equal to (≤). Phrases such as more than and exceeds mean strictly greater than (>), while less than and below mean strictly less than (<).

Choose a variable and state what it represents before writing symbols. For example, let n be a number. The phrase three more than a number is at most twelve becomes n + 3 ≤ 12. The phrase twice a number is greater than seven becomes 2n > 7. Keep the variable on the side that makes the sentence read naturally; the inequality can be reversed later if needed.

1. Define the variable in words.
2. Identify the comparison phrase and its symbol.
3. Write the inequality exactly as the phrase reads.
4. Check by substituting one value that should satisfy the phrase and one that should not.

Solving with Inverse Operations

Solve an inequality much like an equation: undo addition or subtraction first, then undo multiplication or division. The one extra rule is that multiplying or dividing both sides by a negative number reverses the inequality symbol. For instance, -2x < 8 becomes x > -4 after dividing by -2 and flipping the symbol.

Keep the solution in the form variable compared to a number, such as x ≥ 5 or x < -1. This form makes graphing straightforward because the boundary value and the direction are both visible.

1. Simplify each side if needed.
2. Add or subtract the same value from both sides.
3. Multiply or divide both sides by the same nonzero value; reverse the symbol if that value is negative.
4. Write the final inequality with the variable alone on one side.

Graphing on a Number Line

Draw a horizontal number line with evenly spaced tick marks and label enough values to show the boundary and several solutions on each side. Locate the boundary value from the solved inequality. Use a closed circle when the symbol is ≤ or ≥ because the boundary is included. Use an open circle when the symbol is < or > because the boundary is excluded.

Draw an arrow from the circle toward the values that satisfy the inequality. For x ≥ 5, place a closed circle at 5 and draw the arrow to the right. For x < -1, place an open circle at -1 and draw the arrow to the left. The circle and arrow together communicate the entire solution set.

1. Draw and label a number line covering the boundary value.
2. Mark the boundary with an open or closed circle according to the symbol.
3. Draw an arrow in the direction of the solution set.
4. Test one point under the arrow and one point on the other side to confirm the graph.

Checking Your Work

Substitute a test value from the shaded direction into the original phrase or inequality. If the test value makes the sentence true, the arrow direction is likely correct. Then substitute the boundary value itself: if the circle is closed, the boundary should make the inequality true; if the circle is open, it should make the inequality false.

This two-point check catches the most common errors: a reversed arrow and a wrong circle type. It also confirms that the solved inequality matches the original verbal phrase.

1. Pick a test value inside the arrow direction and substitute it.
2. Pick the boundary value and decide whether it should satisfy the inequality.
3. Compare both results with the circle type and arrow direction.
4. Revise the graph if either check fails.

Worked example

Translate and solve: four less than twice a number is at most ten. Then graph the solution on a number line.

Four practice exercises

Common mistakes

• Using a closed circle for < or >. Use an open circle because the boundary value is not part of the solution set.
• Forgetting to reverse the inequality symbol when dividing by a negative number. Circle the negative divisor before dividing, then flip the symbol immediately and write the reversed inequality.
• Drawing the arrow in the wrong direction after solving. Test one value on each side of the boundary in the original inequality; the arrow must point toward the side that makes it true.
• Mistranslating at least as strictly greater than. At least includes the boundary, so use ≥ rather than >.

Self-check rubric

• Variable is defined in words before symbols are used.
• Inequality correctly matches the comparison phrase, including strict versus inclusive wording.
• Solution is obtained with valid inverse operations and the symbol is reversed only when multiplying or dividing by a negative.
• Number line has evenly spaced, labeled tick marks covering the boundary value.
• Circle type matches the inequality symbol: open for < or >, closed for ≤ or ≥.
• Arrow direction matches the solution set and is confirmed by at least one test value.
• Handwriting and circle style may vary; any clearly drawn circle and arrow following the stated convention is accepted.

Questions and answers