Mathematics · 12 / 23
Inequalities: a region of solutions
High school foundations · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
An inequality compares expressions. Adding the same number preserves order. Multiplying or dividing by a negative number reverses order on the number line. Solutions often form an interval. A ≤ or ≥ boundary is included; a < or > boundary is excluded.
- 01Compare
- 02Transform
- 03Reverse if negative
- 04Interval and test
WORKED EXAMPLE
From principle to reasoning
−2x + 3 > 7 gives −2x > 4, then x < −2. x = −3 gives 9 > 7. −2 produces equality and is excluded.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Solve 3 − x ≤ 5.
One hint
Subtract 3, then divide by −1 and reverse order.
Reveal and understand the solution
−x ≤ 2 gives x ≥ −2: [−2, +∞). At the boundary, 3 − (−2) = 5.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Solve −2x < 6.
A hint to get started
Dividing by a negative reverses the inequality.
Step-by-step solution
- Divide both sides by −2.
- x > −3.
- For example 0 works, whereas −4 does not.
Which step did you miss? Retry without help, then explain why the method works.
Solve (x − 1)(x + 2) ≥ 0.
A hint to get started
Mark the two zeros and inspect each sign.
Step-by-step solution
- The zeros are −2 and 1.
- Both factors share a sign outside the roots.
- Solution: (−∞,−2] ∪ [1,+∞), including the zeros.
Which step did you miss? Retry without help, then explain why the method works.
Solve 1/(x − 2) > 0. Why exclude x = 2?
A hint to get started
A positive numerator needs a positive denominator.
Step-by-step solution
- The domain excludes 2 because division by zero is undefined.
- The quotient is positive for x − 2 > 0.
- Solution: (2,+∞).
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Do not divide by an expression of unknown sign without investigating it.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
Choosing a state saves it in this browser. This is your assessment, not a grade.