Linear Inequalities • Topic 4 of 4

Number Line Representation

A number line pictures an inequality's solution set. Use an open circle at the endpoint for a strict inequality (< or >) and a closed (filled) circle for an inclusive one (≤ or ≥), then shade in the direction of the solutions — right for greater-than, left for less-than. A compound "and" inequality is shown as a shaded segment between two circles. Reading these correctly is essential: the type of circle tells you ≤ versus <, and the shading direction tells you which way the solution runs. The SAT often asks you to match an inequality to its graph.

✅ Solved examples

1. How is x > 3 shown on a number line?
An open circle at 3 with shading to the right.
2. How is x ≤ −1 shown?
A closed circle at −1 with shading to the left.
3. What inequality has a closed circle at 2 shaded right?
Closed circle means inclusive; shaded right means greater: x ≥ 2.
4. How is −2 < x ≤ 3 shown?
Open circle at −2, closed circle at 3, shaded between them.

✏️ Practice — try these, take hints as needed

1. How is x < 5 shown on a number line?
Strict inequality → open circle.
Less than → shade left.
At 5.
Open circle at 5, shaded left.
2. How is x ≥ 0 shown?
Inclusive → closed circle.
Greater → shade right.
At 0.
Closed circle at 0, shaded right.
3. What inequality has an open circle at −4 shaded right?
Open = strict.
Shaded right = greater.
x > −4.
x > −4.
4. What inequality has a closed circle at 6 shaded left?
Closed = inclusive.
Left = less than.
x ≤ 6.
x ≤ 6.
5. How is 1 ≤ x < 4 shown?
Closed at 1, open at 4.
Shade between.
—
Closed circle at 1, open circle at 4, shaded between.

📝 Topic test — 8 questions

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