Inequalities and intervals on a number line
Describe a set of possible values and see why a negative multiplier reverses the order.
Core: represent and interpret inequalities, including number lines. Extended also constructs and solves linear and compound inequalities.
Before you begin
- Order negative numbers on a number line.
- Solve a linear equation with signed coefficients.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which is larger: −2 or −5?
What you will learn
- Represent strict and inclusive inequalities on a number line.
- List permitted integers in an interval.
- Construct and solve linear inequalities in Extended work.
The boundary can be included or excluded
The statement x < 3 permits every real value below three but not three itself. Draw an open circle at three. For x ≤ 3, draw a filled circle because equality is included. The direction of the marked interval shows which other values are permitted; a boundary point alone is not a complete representation.
The combined inequality −2 ≤ x < 3 means both conditions must hold. Its interval includes −2 but excludes three. If x is an integer, the values are −2, −1, 0, 1 and 2. If x is real, values such as −1.5 and 2.99 are also permitted. Read the stated number type before listing values.
Extended: preserve the ordering when solving
Adding or subtracting the same number on both sides preserves the direction. Multiplying or dividing by a positive number also preserves it. For 3x + 2 < 14, subtract two and divide by three to obtain x < 4. Unlike an equation, the result is a set of values rather than a single value.
Multiplying or dividing by a negative number reverses the direction. Since 2 < 5, multiplying by −1 gives −2 > −5. Thus −2x ≥ 6 gives x ≤ −3 after division by −2. The reversal follows from reflecting the number line, not from merely moving a term across a sign.
Extended: combine constraints and interpret the model
For −3 ≤ 2x + 1 < 7, subtract one from all three parts, then divide all three by two. The result is −2 ≤ x < 3. A compound inequality is the overlap of two requirements. If they cannot both hold, the solution set is empty.
A budget often creates an inclusive inequality: $10 plus $3 per visit within $25 gives 10 + 3v ≤ 25. Solving gives v ≤ 5, but the context also requires a non-negative integer count. A strict inequality v < 5 would exclude the exact affordable limit of five visits.
Pause and explain
What are the integer values satisfying −1 ≤ x < 2?
Worked example
Extended: solve 7 − 3x < 16 and represent the answer.
Show the worked solution
- Subtract seven to get −3x < 9.
- Divide by −3 and reverse the direction: x > −3.
- Use an open point at −3 and mark the interval to its right. For x = 0 the original inequality holds.
Answer x > −3
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Represent 1 < x ≤ 5 and list its integer solutions.
Give me a hint
Check each boundary's equality sign separately.
Compare my reasoning
- Draw an open circle at one and a filled circle at five.
- Mark the interval between them.
- The integer values are 2, 3, 4 and 5.
Open at 1, filled at 5; integers 2, 3, 4, 5
Look for these in your work
- Used different endpoint conventions correctly.
- Applied the integer restriction.
Extended: solve −4 ≤ 3x + 2 < 11.
Give me a hint
Apply each operation to all three parts.
Compare my reasoning
- Subtract two: −6 ≤ 3x < 9.
- Divide by positive three, keeping both directions.
- The interval is −2 ≤ x < 3; −2 is included and three is excluded.
−2 ≤ x < 3
Look for these in your work
- Transformed all parts of the compound inequality.
- Kept strict and inclusive boundaries distinct.
Extended: a museum costs $8 entry plus $5 for each workshop. With at most $31 available, how many workshops can you attend?
Give me a hint
An affordable cost may equal the budget; the workshop count is an integer.
Compare my reasoning
- Write 8 + 5w ≤ 31, with w a non-negative integer.
- Then 5w ≤ 23, so w ≤ 4.6.
- The maximum integer count is four; five workshops would cost $33.
At most 4 workshops
Look for these in your work
- Used an inclusive budget constraint.
- Rounded down for a whole-number count and checked the next count.
Common mistakes
- Using a filled point for a strict boundary.
- Failing to reverse an inequality after division by a negative.
- Reporting a fractional count in a whole-number context.
What can you explain now?
List the integers satisfying −3 < x ≤ 1.
Compare with the explanation
−2, −1, 0, 1
Exclude −3, include one, and keep only integers in between.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Draw two intervals with different endpoint conventions. Extended: solve a negative-coefficient inequality and check a value on each side of the boundary.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.