Examples of inequalities in life
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used all the time in the world around us—we just have to know where to look. Figuring out how to interpret the language of inequalities is an important step toward learning how to solve them in everyday contexts.
Listening to Language
You are confronted with mathematical inequalities almost every day, but you may not notice them because they are so familiar. Think about the following situations: speed limits on the highway, minimum payments on credit card bills, number of text messages you can send each month from your cell phone, and the amount of time it will take to get from home to school. All of these can be represented as mathematical inequalities. And, in fact, you use mathematical thinking as you consider these situations on a day-to-day basis.
Situation
Mathematical Inequality
Speed limit
Legal speed on the highway ≤ 65 miles per hour
Credit card
Monthly payment ≥ 10% of your balance in that billing cycle
Text messaging
Allowable number of text messages per month ≤ 250
Travel time
Time needed to walk from home to school ≥ 18 minutes
When we talk about these situations, we often refer to limits, such as “the speed limit is 65 miles per hour” or “I have a limit of 250 text messages per month.” However, we don’t have to travel at exactly 65 miles per hour on the highway, or send and receive precisely 250 test messages per month—the limit only establishes a boundary for what is allowable. Thinking about these situations as inequalities provides a fuller picture of what is possible.
Consider this problem:
An 18-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds. The cab (front) of the truck weighs 20,000 pounds, and the trailer (back) of the truck weighs 12,000 pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge?
This problem offers us an upper limit—65,000 pounds—but we are interested in finding out the full range of possibilities for the weight of the truck’s cargo. We can represent the situation using the following inequality, where c is the weight (in pounds) of the truck’s cargo:
cab weight
+
trailer weight
+
cargo weight
≤
allowable weight
20,000
+
12,000
+
c
≤
65,000
Solving this inequality for c, we find that c ≤ 33,000. This means that the weight of the cargo in the truck can be anywhere between 0 pounds and 33,000 pounds and the truck will be allowed to cross the bridge.
20,000 + 12,000 + c
≤
65,000
20,000 + 12,000 + c – 32,000
≤
65,000 – 32,000
c
≤
33,000
Understanding Context
When you are solving or building these inequalities, it is important to know which inequality symbol you should use. Watch for certain phrases that will tip you off:
Phrase
Inequality
“a is more than b”
a > b
“a is at least b”
a ≥ b
“a is less than b”
a < b
“a is at most b;” or
“a is no more than b”
a ≤ b
Many problems, though, will not explicitly use words like “at least” or “is less than.” So how do you figure out which symbol is appropriate in a given situation?
The key is to think about the context of the problem, and to relate the context to one of the situations listed in the table. Context refers to the real-life situation is which the problem takes place.
So, for example, think back to the truck problem that we solved above. The maximum weight allowed on this bridge was 65,000 pounds. We can also think of this relationship using the language of inequalities used in the table: the total weight of the cab, trailer, and cargo had to be no more than65,000. Once we have identified the relationship between the two quantities we can put in the appropriate symbol.
Listening to Language
You are confronted with mathematical inequalities almost every day, but you may not notice them because they are so familiar. Think about the following situations: speed limits on the highway, minimum payments on credit card bills, number of text messages you can send each month from your cell phone, and the amount of time it will take to get from home to school. All of these can be represented as mathematical inequalities. And, in fact, you use mathematical thinking as you consider these situations on a day-to-day basis.
Situation
Mathematical Inequality
Speed limit
Legal speed on the highway ≤ 65 miles per hour
Credit card
Monthly payment ≥ 10% of your balance in that billing cycle
Text messaging
Allowable number of text messages per month ≤ 250
Travel time
Time needed to walk from home to school ≥ 18 minutes
When we talk about these situations, we often refer to limits, such as “the speed limit is 65 miles per hour” or “I have a limit of 250 text messages per month.” However, we don’t have to travel at exactly 65 miles per hour on the highway, or send and receive precisely 250 test messages per month—the limit only establishes a boundary for what is allowable. Thinking about these situations as inequalities provides a fuller picture of what is possible.
Consider this problem:
An 18-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds. The cab (front) of the truck weighs 20,000 pounds, and the trailer (back) of the truck weighs 12,000 pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge?
This problem offers us an upper limit—65,000 pounds—but we are interested in finding out the full range of possibilities for the weight of the truck’s cargo. We can represent the situation using the following inequality, where c is the weight (in pounds) of the truck’s cargo:
cab weight
+
trailer weight
+
cargo weight
≤
allowable weight
20,000
+
12,000
+
c
≤
65,000
Solving this inequality for c, we find that c ≤ 33,000. This means that the weight of the cargo in the truck can be anywhere between 0 pounds and 33,000 pounds and the truck will be allowed to cross the bridge.
20,000 + 12,000 + c
≤
65,000
20,000 + 12,000 + c – 32,000
≤
65,000 – 32,000
c
≤
33,000
Understanding Context
When you are solving or building these inequalities, it is important to know which inequality symbol you should use. Watch for certain phrases that will tip you off:
Phrase
Inequality
“a is more than b”
a > b
“a is at least b”
a ≥ b
“a is less than b”
a < b
“a is at most b;” or
“a is no more than b”
a ≤ b
Many problems, though, will not explicitly use words like “at least” or “is less than.” So how do you figure out which symbol is appropriate in a given situation?
The key is to think about the context of the problem, and to relate the context to one of the situations listed in the table. Context refers to the real-life situation is which the problem takes place.
So, for example, think back to the truck problem that we solved above. The maximum weight allowed on this bridge was 65,000 pounds. We can also think of this relationship using the language of inequalities used in the table: the total weight of the cab, trailer, and cargo had to be no more than65,000. Once we have identified the relationship between the two quantities we can put in the appropriate symbol.
devikrishna60:
Wth....you typed all these?
So fast??
Thankyou so much
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