Inequality Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inequality Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Understanding that < and > describe ordering relationships—one quantity is strictly smaller or larger than the other.

If 5 < 7, then 5 is somewhere to the left of 7 on the number line.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An inequality describes a whole range of valid values, not a single answer like an equation does.

Common stuck point: Multiplying by negative reverses the inequality: if x > 3, then -x < -3.

Sense of Study hint: Plot the boundary value on a number line, then shade the direction that satisfies the inequality to see all solutions.

Worked Examples

Example 1

easy
Solve \(x + 3 < 10\) and graph the solution on a number line.

Solution

  1. 1
    Subtract 3 from both sides: \(x < 7\).
  2. 2
    Solution: all numbers less than 7.
  3. 3
    Graph: open circle at 7, arrow pointing left.
  4. 4
    Example values: \(x = 6, 5, 0, -1, \ldots\)

Answer

\(x < 7\)
Inequalities are solved like equations but with a direction. The solution is a range of values, not a single point. Open circle means 7 is not included.

Example 2

medium
Solve \(-2x \geq 8\) and explain the direction flip when multiplying by a negative.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve \(2x - 4 > 6\).

Example 2

medium
A student must score at least 70 to pass. They have scores 65, 72, 80. Which scores pass? Write as inequalities.
← Back to Inequality Intuition Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

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