Inequalities path · I01

Before this lesson: Arithmetic and exact calculation, Equations and quadratic tools

Your goal: Transform inequalities while preserving equivalent conditions.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Solving Inequalities: Rules and Sign Changes: the key idea

Inequalities compare ordered real quantities. Adding the same number preserves order; multiplying or dividing by a positive number preserves it, and by a negative number reverses it. Multiplying by an expression of unknown sign requires cases. A square is nonnegative, but its square root is defined as nonnegative. To solve a quadratic inequality, factor where possible and examine the sign on intervals separated by its real roots. Include a boundary only when the original comparison permits equality and all expressions are defined.

A worked example

Solve (x−1)(x−4)≤0 over the reals.

The roots are 1 and 4. Below 1 both factors are negative, giving a positive product. Between 1 and 4 the factors have opposite signs, giving a negative product. Above 4 both are positive. The endpoints give zero and are allowed, so 1≤x≤4.

Your turn: change one thing

Solve 3−2x≥9.

Try this on paper before opening the explanation.

Compare your reasoning

Subtract 3 to get −2x≥6. Dividing by −2 reverses the sign, giving x≤−3. The boundary x=−3 checks.

Pause and check

A trap to avoid: Multiplying by an unknown sign or forgetting a zero boundary.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

For positive a<b, prove 1/a>1/b.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Compute 1/a−1/b=(b−a)/(ab). The numerator b−a is positive and the denominator ab is positive. Thus the difference is positive, which proves the reversed order of reciprocals under these assumptions.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. If x<y, which statement always follows?

Foundation

  1. x²<y²
  2. 1/x<1/y
  3. x+3<y+3
  4. −x<−y
Hint

Adding the same number preserves order.

Answer and reasoning

x+3<y+3. Translation preserves order; the other operations need sign and domain conditions.

2. Divide −2x<6 by −2. What results?

Foundation

  1. x<−3
  2. x>3
  3. x<3
  4. x>−3
Hint

Reverse the comparison on division by a negative.

Answer and reasoning

x>−3. Both dividing by −2 and reversing < give x>−3.

3. Solve (x−2)(x−5)<0.

Core

  1. 2<x<5
  2. x<2 or x>5
  3. 2≤x≤5
  4. x=2 or x=5
Hint

The product is negative between its simple roots.

Answer and reasoning

2<x<5. Between 2 and 5 the factors have opposite signs. The strict inequality excludes both endpoints.

4. What is the minimum of (x−3)²+2 over real x?

Core

  1. 1
  2. 3
  3. 2
  4. 0
Hint

A square is at least zero.

Answer and reasoning

2. The minimum occurs when x=3 and the square vanishes.

5. You want to multiply an inequality by x. Which information matters?

Stretch

  1. Only whether x is rational
  2. Whether x is positive, negative or zero
  3. Whether x has two digits
  4. Only whether x is an integer
Hint

The direction depends on sign.

Answer and reasoning

Whether x is positive, negative or zero. Positive and negative multipliers have different effects, and zero can destroy equivalence.

6. For 0<a<b, which inequality is correct?

Stretch

  1. a²>b²
  2. 1/a>1/b
  3. 1/a<1/b
  4. 1/a=1/b
Hint

Write the reciprocal difference.

Answer and reasoning

1/a>1/b. (1/a)−(1/b)=(b−a)/(ab)>0.

7. Multiplying an inequality by a negative number does what?

Foundation

  1. Reverses its direction
  2. Keeps its direction
  3. Makes both sides zero
  4. Requires integers
Hint

Test 1<2 after multiplication by −1.

Answer and reasoning

Reverses its direction. −1>−2, so the comparison reverses.

8. Solve −3x<12.

Core

  1. x<4
  2. x>−4
  3. x<−4
  4. x>4
Hint

Divide by a negative number.

Answer and reasoning

x>−4. Division by −3 reverses the inequality to x>−4.

9. Can one multiply both sides by an expression of unknown sign without cases?

Stretch

  1. Only if the expression is written first
  2. No
  3. Yes always
  4. Only for rational numbers
Hint

Its sign controls the direction.

Answer and reasoning

No. Separate positive, negative and zero cases as needed.

Choose your next step

Continue to Weierstrass product inequalities. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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