Problem 7a
Answer: [tex]18 \le y < 300[/tex]
where y is the amount of tax paid in dollars
So y is between 18 and 300; including 18 but excluding 300.
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Explanation:
x = annual income (in dollars)
y = amount of tax you have to pay (in dollars)
If x is between 600 and 10,000 then y is 3% of that, so y = 0.03*x is the rule for the first tax bracket.
If someone's income is x = 600 dollars, then they have to pay y = 0.03*600 = 18 dollars. This is the "floor" or lowest amount of tax that must be paid. We can't go any lower.
On the other side of the spectrum for this bracket, the highest income possible is x = 10,000 so 3% of this is y = 0.03*x = 0.03*10000 = 300 dollars. This is the "ceiling" or highest amount of tax paid for this bracket.
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Problem 7b
Answer: Annual income is over $10,000
Possible annual incomes can be expressed by this compound inequality [tex]12000 \le x \le 19000 [/tex] meaning the incomes could be between $12,000 and $19,000 (inclusive of both endpoints).
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Explanation:
From part A above, we found that the highest paid tax in the first bracket was $300 (well $299.99 because the exact amount 300 wasn't included). The lowest amount 400 is well above the ceiling mentioned, so we must be in the second tax bracket.
Here's how we compute the second tax bracket: we take 3% of 10,000 to get $300 and then we add on 5% of the remaining income over $10,000. The equation that represents this is y = 0.05*x + 300 where x is the extra amount of money over $10,000. For example, if I said x = 20, then we are 20 dollars over $10,000
Plug in y = 400 and let's solve for x to get
y = 0.05*x + 300
400 = 0.05*x + 300
0.05x = 100
x = 100/0.05
x = 2000
Meaning that a person making an extra $2000 on top of the $10,000 already taxed earns a total of $12,000 annually
Repeat for y = 50
y = 0.05*x + 300
750 = 0.05*x + 300
0.05x = 450
x = 450/0.05
x = 9000
So the annual income is 10,000+x = 10,000+9,000 = 19,000
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Problem 7c
Answer: Anyone earning over $20,000 will pay less in tax at the 4% tax rate.
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Explanation:
y = original tax rate for the second bracket (3% of first 10,000 plus 5% extra)
y = 5% of (extra amount over $10,000) + (3% of 10,000)
y = 5% of (x - 10000) + (3% of 10,000)
y = 0.05*(x-10000) + 300
z = 4% of annual income
z = 0.04*x
We want to see what x values will make z smaller than y
z < y
0.04x < 0.05*(x-10000) + 300
0.04x < 0.05*x-0.05*10000 + 300
0.04x < 0.05*x-500 + 300
0.04x < 0.05*x-200
0.04x-0.05*x < -200
-0.01*x < -200
x > -200/(-0.01) ......... note the inequality sign flips
x > 20000
If you earn more than $20,000 then the original rule of "3% plus the first $10,000 plus 5% of any extra income" will charge more tax. So for higher earners, they will want the flat tax rate of 4%
