HK Math - Algebra
Inequalities
Work from inequality signs to number-line solution sets, then use the properties of inequalities to solve linear inequalities and word problems.
Flow Of Ideas
Start by reading inequality signs correctly. Then represent solution sets on a number line. After that, use the transitive, additive, and multiplicative properties to solve linear inequalities in one unknown. The same solving skills are then used in word problems.
1. Inequality Signs
2. Number Lines
3. Basic Properties
4. Linear Inequalities
5. Problem Solving
1. Inequality Signs
- Understand the meaning of >, <, ≥, and ≤.
- Read inequalities from left to right.
Key Signs
- > : Greater than (e.g., x > 5)
- < : Less than (e.g., x < 3)
- ≥ : Greater than or equal to
- ≤ : Less than or equal to
2. Number Lines
- Use hollow circles for > and <.
- Use solid dots for ≥ and ≤.
- Draw arrows pointing to the correct direction based on the sign.
Graphical Representation
When graphing an inequality on a number line, we use different circle types to indicate whether the endpoint is included in the solution set:
- Hollow Circle (○): Used for > and <. It indicates that the value itself is not part of the solution.
- Solid Dot (●): Used for ≥ and ≤. It indicates that the value is part of the solution.
3. Basic Properties
- Understand the transitive property: if a > b and b > c, then a > c.
- Understand the additive property: if a > b, then a + c > b + c.
- Understand the multiplicative property: if a > b and c > 0, then ac > bc.
- Understand the negative rule: multiplying or dividing by a negative number reverses the inequality sign.
Crucial Rule: Negative Multiplication/Division
When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
-3x > 12
x < -4
4. Linear Inequalities
- Solve linear inequalities in one unknown by isolating the variable.
- Represent the solution set on a number line.
Example 1
4x - 7 > 33
4x > 40
x > 10
Example 2
15 - 6x ≥ 3
-6x ≥ -12
x ≤ 2
5. Problem Solving
- Define the unknown variable.
- Set up an inequality based on the problem description.
- Solve the inequality to find the answer.
- Check if the final answer needs to be an integer (e.g., number of people or items).
Word Problem Example
A Peak tram ticket costs $18. A teacher wants to buy tickets for a group of students and has a budget of at most $500. What is the maximum number of tickets she can buy?
Let n be the number of tickets.
18n ≤ 500
n ≤ 27.77...
Since n must be a whole number, the maximum number of tickets is 27.
Practice Checks
Compare the fractions in the sequence 2/3, 3/5, 4/7, 5/9... What do you find? Can you explain why? Generalize it to the n-th term.Check answer
Let's compare the first few terms by finding common denominators:
2/3 = 10/15 and 3/5 = 9/15. Since 10 > 9, we have 2/3 > 3/5.
3/5 = 21/35 and 4/7 = 20/35. Since 21 > 20, we have 3/5 > 4/7.
We find that the sequence is strictly decreasing. Why? Let's generalize to the n-th term. The numerators are 2, 3, 4, 5... which is n+1. The denominators are 3, 5, 7, 9... which is 2n+1. So the general form for the n-th term is (n+1)/(2n+1).
To prove it is decreasing, compare the n-th term and the (n+1)-th term by subtracting them:
(n+1)/(2n+1) - (n+2)/(2n+3)
= [(n+1)(2n+3) - (n+2)(2n+1)] / [(2n+1)(2n+3)]
= [(2n² + 5n + 3) - (2n² + 5n + 2)] / [(2n+1)(2n+3)]
= 1 / [(2n+1)(2n+3)]
Since n ≥ 1, the denominator is positive, so the result is > 0. This proves that each term is greater than the next term!
Sugar and water problem: Consider a sugar solution where a is the amount of sugar and b is the total volume (so concentration is a/b). If we add x amount of pure sugar, prove that the new concentration is higher, meaning (a+x)/(b+x) > a/b, assuming b > a > 0 and x > 0.Check answer
We want to compare (a+x)/(b+x) and a/b.
Let's subtract them: (a+x)/(b+x) - a/b
= [b(a+x) - a(b+x)] / [b(b+x)]
= (ab + bx - ab - ax) / [b(b+x)]
= x(b - a) / [b(b+x)]
Since b > a, (b - a) is positive. Since x > 0 and b > 0, the denominator is positive.
Thus, the result is > 0, which means (a+x)/(b+x) > a/b.
In real life, adding pure sugar to a sugar-water mixture makes it sweeter!
Represent x > -2 on a number line. Should the endpoint be hollow or solid?Check answer
Use a hollow dot at -2, then draw the arrow to the right. The value -2 is not included because the sign is >.
Solve 5 - 2x ≤ 17. What happens to the inequality sign?Check answer
Subtract 5 from both sides to get -2x ≤ 12. Divide both sides by -2, reverse the sign, and get x ≥ -6.