Content = 1.06 Quiz: Inequalities Matt is five years older than twice his cousin Andy’s age. The sum of their ages is less than 35. Let x represent Andy’s age. Which inequality represents Andy’s possible age
The Correct Answer and Explanation is:
Step 1: Define the variable
Let:
- xxx = Andy’s age (in years)
Step 2: Translate the word problem into expressions
Matt is five years older than twice Andy’s age.
That means Matt’s age can be written as:Matt’s age=2x+5\text{Matt’s age} = 2x + 5Matt’s age=2x+5
The sum of their ages is less than 35, so:x+(2x+5)<35x + (2x + 5) < 35x+(2x+5)<35
Step 3: Simplify the inequality
Combine like terms on the left-hand side:3x+5<353x + 5 < 353x+5<35
Now subtract 5 from both sides:3x<303x < 303x<30
Divide both sides by 3:x<10x < 10x<10
Final Answer:
x<10x < 10x<10
Explanation
This problem requires interpreting a word problem into a mathematical inequality. It involves relationships between two people’s ages and how their combined ages must remain under a certain value.
First, we define the variable xxx to represent Andy’s age. Since Matt is said to be five years older than twice Andy’s age, we represent Matt’s age as 2x+52x + 52x+5. The phrase “twice Andy’s age” means 2x2x2x, and then “five years older” means we add 5 to that.
Next, we look at the condition that the sum of their ages is less than 35. The sum means we add Andy’s age xxx and Matt’s age 2x+52x + 52x+5. So we form the inequality:x+(2x+5)<35x + (2x + 5) < 35x+(2x+5)<35
We combine like terms on the left side to simplify the inequality to 3x+5<353x + 5 < 353x+5<35. Then, we solve it step-by-step by isolating the variable xxx. Subtracting 5 from both sides gives 3x<303x < 303x<30, and dividing both sides by 3 gives the final result x<10x < 10x<10.
This inequality shows that Andy’s age must be less than 10 years old for the total of their ages to remain under 35. The solution is both mathematically valid and logically consistent with the original problem.
