An interval is a stretch of the number line where every value satisfies the inequality. Open circle means the end-point is excluded (< or >); closed circle means it is included (≤ or ≥). You will draw intervals, read them, and list the whole numbers inside them.
This lesson follows reversing an inequality. Once you can solve an inequality correctly, this is how you show the answer clearly.
How do you draw an inequality?
- Mark the end-point value on the line.
- Choose the circle: open for < or >, closed for ≤ or ≥.
- Shade in the direction of the values that work.
- For a double inequality such as a < x ≤ b, draw both circles and join them with one line.
For x > 1 you shade to the right, because the larger values satisfy it. For x ≤ −3 you shade to the left.
Worked example
Show −2 < x ≤ 4 on a number line, then list the integers that satisfy it.
Step 1, left end: the symbol beside −2 is <, so −2 is excluded. Draw an open circle at −2.
Step 2, right end: the symbol beside 4 is ≤, so 4 is included. Draw a closed circle at 4.
Step 3, join them:
-2 -1 0 1 2 3 4
o----------------------------●Step 4, list the integers: the whole numbers greater than −2 and up to 4 are −1, 0, 1, 2, 3 and 4.
Step 5, check the count: that is 6 integers. Test the ends: −2 fails the strict symbol, and 4 passes.
Answer: −2 < x ≤ 4 shown with an open circle at −2, a closed circle at 4, and the integers −1, 0, 1, 2, 3, 4
Some answers have two separate pieces, such as x ≤ −3 or x > 1. Draw two shaded parts with a gap between them. The two pieces are joined with “or”, because a value may satisfy either one.
The mistake to watch for
Mistaken answer: a closed circle at −2 and an open circle at 4, listing the integers −2, −1, 0, 1, 2, 3.
The student swapped the circles, which includes −2 and leaves out 4.
The correction is to read the symbol beside each number separately. In −2 < x ≤ 4, the ”<” touches −2 and the ”≤” touches 4. The symbol decides the circle, and the circle then decides whether that end-point is on your integer list.
Check yourself
1. An open circle is at 3 and the line is shaded to the right. Write the inequality.
Show answer
Open circle means the end-point is excluded, and shading to the right means larger values. So x > 3.
2. List the integers that satisfy −3 ≤ x < 2.
Show answer
−3 is included because of ≤. 2 is excluded because of <. The integers are −3, −2, −1, 0, 1, which is 5 integers.
3. Solve 2x + 1 < 9 and describe the number line.
Show answer
Subtract 1: 2x < 8. Divide by 2 (positive, so no flip): x < 4.
Draw an open circle at 4 and shade to the left.
Check: x = 3 gives 7 < 9, true. x = 4 gives 9 < 9, false.
x < 4
Where this leads next
Next, draw a boundary for a linear inequality to carry the same open and closed idea into two dimensions, where an open circle becomes a dashed line. The non-calculator working trainer supports the arithmetic behind each end-point.
If this lesson made sense but shading in exam conditions still wobbles, online one-to-one Mathematics tuition gives you a teacher who can watch you draw and explain each choice as you make it.