Day 15: Inequalities | Middle Stage Mathematics | Apex Institute of Maths and Sciences

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Day 15: Inequalities | Middle Stage Mathematics | Apex Institute of Maths and Sciences

Day 15: Inequalities | Middle Stage Mathematics

Apex Institute of Maths and Sciences

Level 1: The Quest (Concept) 🔍

In math, things aren’t always exactly equal. An Inequality tells us about the relationship between two values that are not equal. It’s like a balance scale that is tipped to one side!

The Symbols of the Quest:

\( < \) : Less Than (The "L" shape)
\( > \) : Greater Than
\( \leq \) : Less than or equal to
\( \geq \) : Greater than or equal to

Think of the symbol as an alligator’s mouth—it always wants to eat the bigger number!

Level 2: Power-Ups (Tools/Methods) ⚡

The Number Line Rule:
  • Use an Open Circle (\(\circ\)) for \( < \) or \( > \).
  • Use a Closed Circle (\(\bullet\)) for \( \leq \) or \( \geq \).
  • Draw an arrow to the Right for “Greater Than”.
  • Draw an arrow to the Left for “Less Than”.

Solving Trick: Solving an inequality is just like solving an equation \( (x + 2 = 5) \), EXCEPT: If you multiply or divide by a Negative Number, you MUST flip the symbol!

Example: \( -2x < 10 \) becomes \( x > -5 \)

Level 3: Mini-Boss Battles (Daily Life) 🎮

Battle 1: The Theme Park

To ride the “Mega Coaster,” your height \(h\) must be at least 120 cm. This is written as: \( h \geq 120 \). If you are 119 cm, you can’t ride!

Battle 2: The Data Plan

Your mobile data limit is 5GB. To avoid extra charges, your usage \(u\) must be less than or equal to 5. Equation: \( u \leq 5 \).

Level 4: Home Quests (Activities) 🏠

Quest 1: Temperature Watch

Look at the weather forecast with your parents. Find the “Low” and “High” temperatures. Write an inequality for the temperature \(t\) today. (Example: \( 25 \leq t \leq 38 \)).

Quest 2: Measuring Tape Fun

Find 5 items in your house that are shorter than 30 cm. List them and write the inequality \( \text{Length} < 30 \).

Final Boss: Practice Test 🏆

1. EASY Which symbol means “Greater than or equal to”?

Magic Solution: Option D is correct. The horizontal line under the greater than symbol indicates “equal to”.

2. EASY On a number line, what kind of circle do we use for \( x > 5 \)?

Magic Solution: Option A. Since it is strictly “greater than” (not equal), we use an open circle to show 5 is not included.

3. EASY If \( x + 3 < 7 \), then:

Magic Solution: Option B. Subtract 3 from both sides: \( 7 – 3 = 4 \). So, \( x < 4 \).

4. EASY Which inequality represents “A number \(n\) is at most 10”?

Magic Solution: Option C. “At most” means it can be 10 or anything smaller.

5. MODERATE Solve for \( y \): \( 2y – 4 \geq 6 \)

Magic Solution: Option C. Add 4: \( 2y \geq 10 \). Divide by 2: \( y \geq 5 \).

6. MODERATE When graphing \( x \leq -2 \), which way does the arrow point?

Magic Solution: Option B. “Less than” values are always to the left on a standard number line.

7. MODERATE Solve: \( \frac{x}{3} + 1 > 2 \)

Magic Solution: Option A. Subtract 1: \( x/3 > 1 \). Multiply by 3: \( x > 3 \).

8. MODERATE Is \( x = 4 \) a solution to \( 3x – 2 < 10 \)?

Magic Solution: Option B. \( 3(4) – 2 = 10 \). Since 10 is NOT less than 10, it’s False.

9. COMPLEX Solve and choose carefully: \( -4x < 12 \)

Magic Solution: Option D. Divide by -4. Rule: When dividing by a negative, FLIP the symbol! \( x > -3 \).

10. COMPLEX Solve: \( 5 – x \geq 2 \)

Magic Solution: Option B. Subtract 5: \( -x \geq -3 \). Multiply by -1 and flip: \( x \leq 3 \).
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