# INEQUALITIES AND RANGES OF NUMBERS

The symbol < means “is less than,” as in 3 < 4. If x represents the numbers of the months of the year, for example, x <4 means x could be1,2,or3. The symbol < means “is less than or equal to,” so that x < 4 in the above example means x could be 1, 2, 3, or 4.

Similarly, the symbols > and > mean “is greater than” and “ts greater than or equal to,” respectively. For some reason, these latter symbols are rarely used. Instead, the more common approach is to say that x < 3 means that x is less than 3, and 3 < x means that 3 is less than x, or equivalently, that x is equal to or greater than 3.

These symbols let us describe a range of numbers conveniently. For example,

3 < x < 7

means that x is somewhere between 3 and 7, but is not equal to either 3 or 7, while

3 < x < 7

means that x is somewhere between 3 and 7, possibly 7 itself, but not 3.

This notation will prove to be very useful in describing income tax brackets. For example, if y is your income, then the simple table means that if your income is less than \$10,000, your tax is \$25; if your income is \$10,000 or more but less than \$22,500, your tax is \$235; and if your income is \$22,500 or more, your tax is \$1,000. (This is not an actual tax table; I’m just inter­ested in describing ranges of numbers using inequality signs here.)

PROBLEMS 1. Evaluate the following arithmetic expressions: 7-(12-5) 12(14-6)

16 -(3 + 7)

3 (7 – 5)

(12 – 2)(7 + 3) 12 – 2(7 + 3) (12-2)7 + 3 6.2 + 1/3

2. Given that x = 6, y = 2, and г = 3, evaluate the following arithmetic expressions:

(a) x + y + г

(b) г(х – 3)(y + 2) x + 2

(c) – y—3 + 2.25 г – 4

(d) x(x – 1)(x + 2)

3. Refer to the table for the problem set 3:

 T (hours) P (\$) N (number of wallets sold) 0 25.00 6 2 24.00 2 4 23.00 2 6 21.00 4 8 18.00 8 10 14.00 16

This table is for the price (P) of a wallet at a luggage counter over the course of a day set by a store owner who would be getting a shipment of new wallets the following day and wanted to make sure that he had cleared the old wallets out of his inventory before the
new ones came in. T is the time in hours, starting the count from when the store opened. N is the number of wallets sold at the corresponding price (same row in the table).

(a) If we use subscripted notation to refer to the table entries for T and P, what is T1, P3, N4, and T5?

(b) How many wallets were sold at more than \$20 per wallet? How many wallets were sold all together?

(c) If the storekeeper paid \$12 each for these wallets, not counting overhead, did he make or lose money that day? If overhead (rent, electricity, etc.) costs \$100 a day, did the storekeeper make or lose money that day?

4. Let’s extend problem 3 to a 2-day rather than a 1-day wallet sale. On the second day, the storekeeper decides to repeat his change – the – price – every – two – hours strategy, but he drops each price by 10% from its amount on the first day. His sales in each 2-hour period on the second day are 50% higher than they were on the first day.

(a) Create a table just like the table in problem 3, except use numbers for the second day.

(b) Using the same assumption about overhead as in problem 3, how much money did the storekeeper make or lose on the second day?

5. Sketch a histogram based on the table of problem 3 using T as the horizontal axis and N as the vertical axis.

6. The table below shows a business plan to put items on sale. Shown are the original price and the sale price of a list of items. Calculate the percent change of all the items to the nearest percent and put these numbers in the empty percent change locations in the table. Calculate the percent sale numbers (% Sale = 100 – % Change). Then plot a graph of these data, placing the original price on the horizontal axis and the percent sale on the vertical axis.

Original price (\$) Sale price (\$) % Change % Sale

 289.99 217.49 249.99 199.99 127.50 102.00 99.99 84.99 59.79 53.99 37.50 33.75 For x = 0.5 and then again for x = 1.2, fill out the table below:

n (0.25 + x)n (0.25 + x)n

x = 0.5 x =1.2

0

1

2

3

0 (Use a calculator or a spreadsheet for this problem; doing it by hand is a very tedious job.) Can you draw any conclusions about numbers raised to the 0th power and numbers raised to the 1st power?

8. Approximate (round) the following numbers to the nearest dollar: \$12.87, \$22.22, \$53.50, and \$1,719.88.

9. The following table describes a walk I recently took. I started at 1:30 and walked until 3:30:

 Time walked (miles) Distance 1:30 0 1:45 0.25 2:00 0.5 2:15 0.75 2:30 1.00 2:45 1.50 3:00 2.00 3:15 2.50 3:30 3.00

Draw a graph using time walking (in hours) as the horizontal axis and distance walked (in miles) as the vertical axis. What was my average speed for the entire walk? (Don’t forget to specify your units.) Can you estimate the instantaneous speeds near the beginning of the walk and near the end of the walk?

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