Linear inequality in one variable – solution set on a number line

What it shows

Solving ax + b > c follows the same moves as an equation, except that dividing both sides by a negative number reverses the inequality sign. The solution set is a ray from x₀ = (c − b)/a: a filled dot means x₀ is included (≥ or ≤), an open dot that it is not. The Intervals tab names such sets: closed, open, half-open and unbounded intervals, written in interval, inequality and set-builder notation. Combining two intervals by union, intersection, difference or complement shows how a compound inequality with and or or becomes one interval, two pieces, a single number or the empty set.

How to use

In the Solve tab, press Next step to reveal the working, then make Coefficient a negative and predict which way the shading turns; drag the red test point to substitute values. In the Intervals tab, set A and B with the bracket buttons and endpoints, or drag the endpoints on the line and tap them to include or exclude them. Switch Operation between A ∪ B, A ∩ B, A ∖ B and ℝ ∖ A, read the table, then answer the Practice questions.

Parameters you can change

• Coefficient a -10–10
• Coefficient b -20–20
• Right-hand side c -20–20
• Inequality sign ax + b > c, ax + b < c, ax + b ≥ c, ax + b ≤ c
• Screen Solve the inequality, Intervals and operations
• A: left endpoint (−11 means −∞) -11–10
• A: right endpoint (11 means +∞) -10–11
• A: brackets closed [a, b], half-open [a, b), half-open (a, b], open (a, b)
• B: left endpoint (−11 means −∞) -11–10
• B: right endpoint (11 means +∞) -10–11
• B: brackets closed [a, b], half-open [a, b), half-open (a, b], open (a, b)
• Operation union A ∪ B, intersection A ∩ B, difference A ∖ B, difference B ∖ A, complement ℝ ∖ A

Questions to explore

1. Why must the inequality sign be reversed when both sides are divided by a negative a?
2. When a = 0, how can the inequality be true for every x or have no solution?
3. Why is the solution of x ≥ 3 and x < 1 the empty set, while x ≥ 3 or x < 1 gives two intervals?