This module introduces absolute value inequalities.
Here’s one of my favorite problems:
Having seen that the solution to
is
, many students answer this question
. However, this is not only wrong: it is, as discussed above, relatively meaningless. In order to approach this question you have to—you guessed it!—step back and think.
Here are two different, perfectly correct ways to look at this problem.
What numbers work? 4 works. –4 does too. 0 works. 13 doesn’t work. How about –13? No: if
then
, which is not less than 10. By playing with numbers in this way, you should be able to convince yourself that the numbers that work must be
somewhere between –10
and 10. This is one way to approach the answer.
The other way is to think of absolute value as representing
distance from 0 .
and
are both 5 because both numbers are 5 away from 0. In this case,
means “the distance between x and 0 is less than 10”—in other words, you are within 10 units of zero in either direction. Once again, we conclude that the answer must be
between –10 and 10.
It is not necessary to use both of these methods; use whichever method is easier for you to understand.
More complicated absolute value problems should be approached in the same three steps as the equations discussed above: algebraically isolate the absolute value, then
think , then algebraically solve for
. However, as illustrated above, the
think step is a bit more complicated with inequalities than with equations.
Absolute value inequality
Algebraically isolate the absolute value
(don’t forget to switch the inequality when dividing by –3!)
Think!
As always, forget the
in this step. The absolute value of
something is greater than 13. What could the
something be?
We can approach this in two ways, just as the previous absolute value inequality. The first method is trying numbers. We discover that all numbers greater than 13 work (such as 14, 15, 16)—their absolute values are greater than 13. Numbers less than –13 (such as –14,–15,–16) also have absolute values greater than 13. But in-between numbers, such as –12, 0, or 12, do not work.
The other approach is to think of absolute value as representing distance to 0. The distance between
something and 0 is greater than 13. So the
something is more than 13 away from 0—in either direction.
Either way, we conclude that the
something must be anything greater than 13, OR less than –13!
Many students will still resist the
think step, attempting to figure out “the rules” that will always lead from the question to the answer. At first, it seems that memorizing a few rules won’t be too hard: “greater-than problems always lead to
OR answers” and that kind of thing. But those rules will fail you when you hit a problem like the next one.
Absolute value inequality
Algebraically isolate the absolute value
Think!
The absolute value of
something is greater than –3. What could the
something be? 2 works. –2 also works. And 0. And 7. And –10. And...hey! Absolute values are always positive, so the absolute value of anything is greater than –3!
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product