Question

Aaron is earning money by mowing lawns. He charges \(\$ 40\) per lawn. He is trying to save up at least \(\$ 1500\) for a new riding lawnmower. \(\mathrm{He}\) already has \(\$ 420\). Which inequality best represents Aaron's riding lawnmower goal? A. \(420 x+40 \geq 1500\) B. \(40 x+420 \geq 1500\) C. \(420 x+40 \leq 1500\) D. \(40 x+420 \leq 1500\)

Short answer

Option B: \( 40x + 420 \geq 1500 \)

Step-by-step solution

Identify key information

Identify the values given in the problem. Aaron charges \( \$40 \) per lawn and aims to save at least \( \$1500 \). He already has \( \$420 \).

Set up the inequality

Aaron's goal is to save at least \( \$1500 \). This can be expressed as needing at least \( 420 + 40x \) dollars (where \( x \) is the number of lawns mowed) being greater than or equal to \( 1500 \).

Form the inequality

To express this mathematically, we write: \[ 40x + 420 \geq 1500 \]

Verify the options

Compare the formed inequality with the given options. The correct representation is: \[ 40x + 420 \geq 1500 \]

Key concepts

Understanding Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). In the given exercise, we identified key values and variables. Aaron charges \(40 per lawn and already has \)420 saved. The unknown variable here is the number of lawns he needs to mow, which we'll call \( x \).
An algebraic expression for his total earnings from mowing \( x \) lawns would be \( 40x + 420 \). This is crucial for forming the inequality that helps solve the problem.

Solving Inequalities

Inequalities are mathematical statements that compare two expressions. They often use symbols like \( \geq \) (greater than or equal to) and \( \leq \) (less than or equal to).
For Aaron's goal, we need to set up an inequality to see how many lawns he must mow to have at least \(1500.
We start with the algebraic expression for his earnings, \( 40x + 420 \), and we set it greater than or equal to \)1500, like so:
\( 40x + 420 \geq 1500 \).
To solve this inequality, we first subtract $420 from both sides:
\( 40x \geq 1080 \).
Then, we divide both sides by 40 to isolate \( x \):
\( x \geq 27 \).
Therefore, Aaron needs to mow at least 27 lawns to meet his goal.

Mathematical Reasoning

Mathematical reasoning involves logically breaking down a problem and solving it step by step. Here, we used reasoning to identify what Aaron already has and what he needs to meet his goal. We correctly translated a real-world situation into a mathematical inequality.
To break it down:

• Identify Aaron's current savings and earnings per lawn
• Translate these into an algebraic expression
• Formulate the inequality based on his goal
• Solve the inequality step by step

This process helps in making informed decisions and finding the correct mathematical representation, which in this case, is \( 40x + 420 \geq 1500 \). Proper reasoning ensures that the steps are accurately followed to reach the correct solution.