Office Max is offering a $40 rebate on all color printers. Sara is looking at a few different color printers that range in price from $165 to $295. How much can she plan to pay after the rebate?
Part A: Form a compound inequality to represent this problem and solve.
Part B: Can she stay within her budget of $123 for the printer? Why or why not?
Part A:
Let x be the price of the color printer.
The compound inequality is:
165 ≤ x ≤ 295.
After the rebate of $40, the price she would have to pay will be:
x - $40.
So, the compound inequality after the rebate is:
165 - $40 ≤ x - $40 ≤ 295 - $40.
Simplifying the compound inequality:
125 ≤ x - $40 ≤ 255.
Part B:
If Sara has a budget of $123 for the printer, then we can represent this as an inequality:
x ≤ $123.
Comparing the inequality x ≤ $123 with the compound inequality 125 ≤ x - $40 ≤ 255, we can see that x ≤ $123 satisfies the compound inequality.
Therefore, Sara can stay within her budget of $123 for the printer.
Part A: To find out how much Sara can plan to pay after the rebate, we need to subtract the rebate amount from the original price of the printer. Let's say the original price is represented by x. The rebate is $40. So, the equation to represent this problem would be: x - 40.
Now, we know that the price of the printers range from $165 to $295. To form a compound inequality, we need to express this range using inequality symbols. We can do this by setting up two inequalities:
1. x ≥ 165 (Since the price must be greater than or equal to $165)
2. x ≤ 295 (Since the price must be less than or equal to $295)
Combining these two inequalities, we get the compound inequality: 165 ≤ x ≤ 295.
To find out how much Sara can plan to pay after the rebate, we substitute x with the rebate equation: 165 ≤ (x - 40) ≤ 295.
Simplifying the compound inequality, we get: 165 + 40 ≤ x ≤ 295 + 40
This results in the compound inequality: 205 ≤ x ≤ 335.
Part B: To determine if Sara can stay within her budget of $123, we need to check if the price (x) falls within her budget.
From the compound inequality we found in Part A, we know that the price (x) must be between 205 and 335.
Since $123 is less than 205 (the lower bound), Sara would not be able to stay within her budget of $123. She would need to increase her budget to at least $205 in order to purchase a printer and still be eligible for the rebate.
Part A: To represent the problem as a compound inequality, let's assume the price of the color printer is x.
The given rebate is $40, so the amount Sara will pay after the rebate can be calculated by subtracting the rebate from the original price of the printer: x - 40.
Since Sara is considering color printers that range in price from $165 to $295, we can form a compound inequality as follows:
165 ≤ x ≤ 295.
Now, substituting the value x - 40, we have:
165 ≤ x - 40 ≤ 295.
To solve this compound inequality, we need to isolate x in the middle part:
165 + 40 ≤ x ≤ 295 + 40,
205 ≤ x ≤ 335.
Therefore, Sara can expect to pay between $205 and $335 after the rebate.
Part B: Sara's budget for the printer is $123. To determine if she can stay within her budget, we need to check if the price after the rebate, x - 40, is less than or equal to $123.
Using the upper limit from the previous solution, we have:
x - 40 ≤ 123.
Adding 40 to both sides of the inequality, we get:
x ≤ 163.
Therefore, if Sara's budget for the printer is $123, she can stay within her budget because all the prices that satisfy the compound inequality x - 40 ≤ 123 are within the range of prices she is considering.