Think about the differences between equations and inequalities. For what purpose do you use each one? How is solving for them the same and how is it different?

Question

In 1 sentence Create and share a real-world scenario for which you will create an inequality and equation that requires at least two steps to solve. After you have created and shared your own scenario, solve the scenarios shared by two of your peers.
In your response to each post:
Write the solution to both the inequality and the equation.
Explain how the inequality and the equation work together in the scenario
Post 1: Imagine I want to purchase a truck. It costs $9,000. I already have five hundred dollars saved. I get paid $250 a week. How many weeks do i have to work to get the total amount of money for the truck?
Post 2: I am saving money for a new fishing rod and reel. I got a $100 gift card to Bass Pro Shops from my mom. The rod I want costs $200 and the reel costs $300. If I save $50 a week will it ity take greater than or less than 10 weeks to save the money?

Answer 1

My Scenario:

I want to buy a gaming laptop that costs $1,200. I currently have $400 saved, and I save $100 each month. How many months will it take for me to afford the laptop?

Post 1 Response:

In the scenario of purchasing a truck:

1. Equation: To find out how many weeks (\( w \)) it will take to save for the truck:
   \[ 500 + 250w = 9000 \] Subtracting 500 from both sides:
   \[ 250w = 8500
   \]
   Dividing by 250:
   \[ w = 34 \]

2. Inequality: To determine how many weeks it takes for the amount saved to be at least $9,000:
   \[ 500 + 250w \geq 9000 \]
   Solving it leads to:
   \[ 250w \geq 8500 \implies w \geq 34 \]

Explanation: The equation gives the exact number of weeks needed to reach exactly $9,000, which is 34 weeks, while the inequality shows that at least 34 weeks are necessary to meet or exceed that amount. Both express the same concept but in different terms — the equation for a precise answer and the inequality for a range of feasible timeframes.

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Post 2 Response:

In the scenario of saving for a fishing rod and reel:

1. Equation: To determine if savings over 10 weeks can meet the total cost of $500:
   \[ 100 + 50w = 500 \]
   Subtracting 100 from both sides:
   \[ 50w = 400
   \]
   Dividing by 50:
   \[ w = 8 \]

2. Inequality: To check if it will take more than 10 weeks to save up:
   \[ 100 + 50w < 500 \]
   Subtracting 100 gives:
   \[ 50w < 400 \implies w < 8 \]

Explanation: The equation tells us that exactly 8 weeks will allow for the purchase of both items, while the inequality shows that if I only save for 10 weeks, I will have saved too much, which reflects that the savings goal can indeed be met in less than the 10-week threshold. Thus, the inequality confirms that I will achieve my goal in less time than expected.