Apply inequalities to solve everyday problems
Sarah has KSh 12,000 to spend on school supplies. She needs to buy notebooks that cost KSh 300 each and pens that cost KSh 200 each. She needs at least 5 notebooks and at least 10 pens. How many of each can she buy while staying within her budget?
Which inequality represents this situation? Let x be the number of notebooks and y be the number of pens.
Your Progress:
> Greater than
< Less than
≥ Greater than or equal to
≤ Less than or equal to
ax + by ≤ c (or ≥, <, >)
Where a, b, and c are constants, and x and y are variables
1. Graph the boundary line (= instead of inequality)
2. Use solid line for ≤ or ≥
3. Use dashed line for < or >
4. Shade the region that satisfies the inequality
To determine which side to shade:
1. Choose a test point (often the origin)
2. Substitute into the inequality
3. If true, shade the region containing the point
4. If false, shade the opposite region
If you have KSh 5,000 to spend on items costing KSh 500 and KSh 1,000:
500x + 1000y ≤ 5000
Where x = number of KSh 500 items and y = number of KSh 1,000 items
A factory needs to produce at least 100 units:
x + y ≥ 100
Where x = units from machine A and y = units from machine B
If homework takes 2 hours and studying takes 1.5 hours, with 10 hours available:
2x + 1.5y ≤ 10
Where x = homework assignments and y = study sessions
If you need at least 2000 calories from two food sources:
300x + 400y ≥ 2000
Where x = servings of food A (300 cal) and y = servings of food B (400 cal)
1. Look for quantities that can vary or change
2. Assign variables to these quantities (x, y, etc.)
3. Clearly define what each variable represents
"At least" → ≥
"At most" or "No more than" → ≤
"More than" → >
"Less than" → <
"Maximum" → ≤
"Minimum" → ≥
Real-life problems often have multiple constraints:
1. Write each constraint as a separate inequality
2. The solution must satisfy ALL constraints
3. Graph all constraints to find the feasible region
In most real-life situations, variables can't be negative:
x ≥ 0, y ≥ 0
Always consider these constraints even if not explicitly stated