Lesson OVERVIEW of lessons on inequalities and domains of functions

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OVERVIEW of lessons on inequalities and domains of functions


My lessons on inequalities in this site are
    - Solving simple and simplest linear inequalities
    - Solving absolute value inequalities
    - Advanced problems on solving absolute value inequalities
    - Solving systems of linear inequalities in one unknown
    - Solving compound inequalities

    - What number is greater? Comparing magnitude of irrational numbers
    - Arithmetic mean and geometric mean inequality
    - Arithmetic mean and geometric mean inequality - Geometric interpretations
    - Harmonic mean
    - Prove that if a, b, and c are the sides of a triangle, then so are sqrt(a), sqrt(b), and sqrt(c)

    - Solving problems on quadratic inequalities
    - Solving inequalities for high degree polynomials factored into a product of linear binomials
    - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials
    - Solving inequalities for rational functions with non-zero right side
    - Another way solving inequalities for rational functions with non-zero right side

    - Advanced problems on inequalities
    - Challenging problems on inequalities
    - Solving systems of inequalities in two unknowns graphically in a coordinate plane
    - Solving word problems on inequalities
    - Proving inequalities
    - Math circle level problems on inequalities
    - Math Olympiad level problems on inequalities
    - Entertainment problems on inequalities
under the topic  Inequalities, trichotomy of the section  Algebra-I.

My lessons on domains of functions are
    - Domain of a function which is a quadratic polynomial under the square root operator
    - Domain of a function which is a high degree polynomial under the square root operator
    - Domain of a function which is the square root of a rational function.
under the topic  Functions, Domain  of the section  Algebra-I.

Below is the same list of lessons on inequalities with solved problems


Solving simple and simplest linear inequalities

    1)  x-6 > 2.      2)  2x + 4 > -14.      3)    -4x - 6 > 18.    4)  4x-6 < 2x + 3.      5)  2*(3x-2) < 4*(x + 1).      6)  4*(2-3x) < 3*(x + 1).

    7)  Solve an inequality   3x - 1 > x+5   and graph the solution set.


Solving absolute value inequalities

    1)  |x| <= 3.     2)  |x| > 7.        3)  |x-2| <= 3.        4)  |x-3| >= 2.        5)  |x-5| -2 <= 7.

   6)  |7x-3| <= 18.        7)  |2x - 5| >= 1.        8)  4|2x+3|-7 < 9.        9)  6|9b-1|-4 > 2.        10)  5-6|8n+8|>-91.


Advanced problems on solving absolute value inequalities

        Problem 1.  Solve an inequality   |2*|x-3|-7| < 5.

        Problem 2.  Solve an inequality   |3*|x-5|-8| > 5.

        Problem 3.  Solve an inequality   |h+3| + |h-3| < 6.

        Problem 4.  Solve an inequality   2|4-3x|-3|2x+1| < 7.


Solving systems of linear inequalities in one unknown

        Problem 1.  Solve the system of inequalities   2x >-10  and  9x < 18.

        Problem 2.  Solve the system of inequalities   2x > 10  OR  9x < -18.

        Problem 3.      b <= 2a - 1
                               8 > a - b
                           According to the system of inequalities above,  which of the following could be a value of a?
                               (A) -12;     (B) -8;     (C) -7;     (D) -6.

        Problem 4.  Solve this system of inequalities and write interval notation for the solution set.  Then graph the solution set.
                               2x - 10 < - 1.4   or   2x-10 > 1.4.


Solving compound inequalities

        Problem 1.  Solve a compound inequality   10 < 2x+4 < 16.

        Problem 2.  Solve an inequality   -9 < 2x+7 <= 19   and write the solution set using interval notation.

        Problem 3.  Solve a compound inequality   -13 <= 5 - 2x <= 25.

        Problem 4.  Solve a compound inequality   3y < 5-2y < 7+y.

        Problem 5.  Solve a compound inequality   1.99 < 1%2Fx < 2.01.

        Problem 6.  Solve a compound inequality   0 < 1%2F%289x-27%29 < 1/9.


What number is greater? Comparing magnitude of irrational numbers

        Problem 1.  What number is greater,  root%283%2C3%29  or  sqrt%282%29 ?

        Problem 2.  What number is greater,  root%285%2C5%29  or  sqrt%282%29 ?


Arithmetic mean and geometric mean inequality
The Arithmetic mean - Geometric mean inequality is a famous,  classic and basic Theorem on inequalities.  It states that . . .
Geometric mean of two real positive numbers is lesser than or equal to their arithmetic mean.
Geometric mean of two real positive unequal numbers is less than their arithmetic mean.
Geometric mean of two real positive numbers is equal to their arithmetic mean if and only if the two numbers are equal.

Arithmetic mean and geometric mean inequality - Geometric interpretations
Geometric mean is interpreted
1)  as the height of a right angled triangle drawn to its hypotenuse,     and
2)  as the chord in a circle, drawn perpendicularly to its diameter.

Harmonic mean

        Problem 1.  For   0 < a < b,   let  h  be defined by   1%2Fh = %281%2F2%29%2A%281%2Fa%29+%2B+%281%2Fb%29%29.   Show that   a < h < b.
                           The number  h  is called the harmonic mean of  a  and  b.


Prove that if a, b, and c are the sides of a triangle, then so are sqrt(a), sqrt(b), and sqrt(c)

        Problem 1.  Prove that if  "a",  "b",  and  "c" are the sides of a triangle,  then so are  sqrt%28a%29,  sqrt%28b%29,  and  sqrt%28c%29.

        Problem 2.  Prove that if  "a",  "b",  and  "c"  are the sides of a triangle,  then  so  are   root%28n%2Ca%29,   root%28n%2Cb%29,   and   root%28n%2Cc%29   for any integer positive index  n >= 2.

        Problem 3.  Let   f:   R+  --->  R+   be a monotonically risen one-to-one function such that  f(a) + f(b) > f(a+b)  for all real positive numbers  "a"  and  "b".
                           Prove that if  "a",  "b",  and  "c"  are the sides of a triangle,  then so are  f(a),  f(b),  and  f(c).


Solving problems on quadratic inequalities

    1)  x%5E2-4 >= 0;    2)  -x%5E2%2B4 >= 0;    3)  %28x-1%29%2A%28x-2%29 >= 0;    4)  -%28x-2%29%2A%28x-3%29 >= 0;    5)  x%5E2+-+4x+%2B+3 >= 0;    6)  -x%5E2+%2B+2x+%2B+3 >= 0;    7)  x%5E2+%2B+1 >= 0.


Solving inequalities for high degree polynomials factored into a product of linear binomials

    1)  %28x-1%29%2A%28x-2%29%2A%28x-3%29 >= 0   and   2)  %28x-1%29%2A%28x-2%29%2A%28x-3%29%2A%28x-4%29 >= 0.


Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials

    1)  1%2F%28x-1%29 > 0;     2)  %28x-1%29%2F%28x-2%29 >= 0;     3)  %28x-1%29%2F%28x-2%29 < 0;     4)  %28x-2%29%2F%28%28x-1%29%2A%28x-3%29%29 > 0;     5)  %28%28x-1%29%2A%28x-3%29%29%2F%28%28x%2B1%29%2A%28x-2%29%29 < 0;   6)    %28x%5E2-2%29%2F%28x%5E2-4%29 > 0.

Solving inequalities for rational functions with non-zero right side

    1)  x%2F%28x-6%29 < 2;     2)  %28x-5%29%2F%283x%29 < 3;     3)  %283x%2B2%29%2F%28x%2B1%29 > 4;     4)  %283x%2B1%29%2F%28x-2%29 > 2.


Another way solving inequalities for rational functions with non-zero right side

        Problem 1.  Solve an inequality   %282x-5%29%2F%285+%2B+2x%29 <= 0.

        Problem 2.  The numerator of the fraction is  x,  while the denominator is  5  more the numerator.
                           If the value of the fraction is less than  2/5,  find the value of  x.

        Problem 3.  Solve an inequality   4%2F%282-x%29 <= 6%2F%284%2Bx%29.


Advanced problems on inequalities

        Problem 1.  Without using a calculator or technology,  determine which number is greater:   e%5Epi  or  pi%5Ee ?

        Problem 2.  Without using a calculator or technology,  determine which number is greater:   13%5E31  or  31%5E13 ?

        Problem 3.  Without using a calculator or technology,  determine which number is greater,  sqrt%282017%29%5E%28sqrt%282016%29%29  or  sqrt%282016%29%5E%28sqrt%282017%29%29 ?

        Problem 4.  Which number is greater:   log%284%2C%2817%29%29   or   log%285%2C%2824%29%29 ?   Solve without using a calculator.

        Problem 5.  Which number is greater:   log%282%2C%283%29%29   or   log%283%2C%285%29%29 ?   Solve without using a calculator.

        Problem 6.  Which number is greater:   log%282%2C%283%29%29   or   log%283%2C%288%29%29 ?   Solve without using a calculator.


Challenging problems on inequalities

        Problem 1.  If   x+y+z=16,   find the maximum value of   (x-3)(y-5)(z-2),  given that   (x-3) > 0, (y-5) > 0, (z-2) > 0.

        Problem 2.  Determine the set of all real  x  satisfying   (x^2 - 5x + 3)^2 < 9 + 6x + x^2.


Solving systems of inequalities in two unknowns graphically in a coordinate plane

        Problem 1.  Solve an inequality graphically   2y > -x - 2.

        Problem 2.  Solve this system of inequalities by graphing   y <= 3x-2,  y >= 1-x.

        Problem 3.  How to sketch the following inequalities on the graph ?
                           Give a wording description of sketches for each separate inequality.
                               (A)   y < -4;
                               (B)   x >= 5;
                               (C)   y-7 <= 0.

        Problem 4.  Solve the system of inequalities graphically   2x + y >= 8,  x + 2y >= 8,  x + y <= 6.

        Problem 5.  Solve the system of inequalities graphically   2x +3y <= 6,  x + 4y <= 4,  x >= 0,  y >= 0.

        Problem 6.  Solve the system of inequalities graphically   x - 2y >= 0,  2x - y <= -2,  x >= 0,  y >= 0.

        Problem 7.  Graph the feasible region for the follow system of inequalities
                           by drawing a polygon around the feasible region.
                               x + 5y <= 40
                               7x + 5y <= 70
                               x >= 0,  y >= 0

        Problem 8.  How many ordered pairs of positive integer numbers  (x,y)  are there such that their sum is  45
                           and the difference of the first and the second numbers is less than  10 ?

        Problem 9.  How many ordered pairs of positive integer numbers  (x,y)  are there such that their sum is  45
                           and the distance between them in the number line is less than  10 ?


Solving word problems on inequalities

        Problem 1.  The sum of two whole numbers is  45  and the distance between them in a coordinate plane is less than  10.
                           What is the number of all such possible pairs ?

        Problem 2.  A test is divided into  4  sets of problems with the same number of problems in each set.
                           Alice correctly solves  35  problems.  How many problems are on the test if  Alice solved more than  60  percent of all problems,
                           but less than  65 percent of all problems?

        Problem 3.  You have been asked to contribute to a service project benefiting a local organization.  You plan to buy a combination
                           of  $5  dollar and  $10  gift cards to donate.  You plan to buy exactly  20  gift cards and want to spend
                           between  $150  and  $160.  What are the possible combinations of $5 and $10 gift cards you can purchase?

        Problem 4.  Britney wants to bake at most  10  loaves of bread for a bake sale.
                           She wants to make banana bread that sells for  Php 50  each loaf and carrot bread for  Php 60  each and make at least  Php 580  sales.
                           Write a system of inequalities for the given situation and solve it graphically.

        Problem 5.  Jose bought  4  identical markers and  3  identical pencils.  He paid at most  $25  for his purchase.
                           Each marker is  $3.00  more expensive than each pencil.
                           What is the price of the marker and the price of the pencil ?

        Problem 6.  For which values of  k  do the lines   x+ky = 4  and   2x-3y = 6   intersect in the  3rd quadrant?

        Problem 7.  Larry plans to make at least  10  pounds of a snack mix that will consist of almonds and dried fruit.
                           If the wants the snack mix to be at least  60%  almonds by weight,  which system of inequalities
                           represents the constraints on the number of pounds of almonds,  a,  and the number of pounds of dried fruit,  f ?

        Problem 8.  A bag contains  100  marbles,  some red,  the rest blue.  If there are no more than one and a half times
                           as many red marbles as blue ones in the bag
                               (a)   at most how many red marbles are in the bag?
                               (b)   at least how many blue ones are in the bag?

        Problem 9.  The intensity  I  (in candlepower)  of a certain light source obeys the equation  I = 900/(x^2),
                           where  x  is the distance  (in meters)  from the light.  Over what range of distances can an object
                           be placed from this light source so that the range of intensity of light is from  1600  to  3600 candlepower,  inclusive?


Proving inequalities

        Problem 1.  Show that in general,  if  "a   and  "b"  are positive and   a < b,  then   a%2F%28ax+%2B+y%29 < b%2F%28bx+%2B+y%29,
                           where  x  and  y  can be any positive real numbers.

        Problem 2.  Let  "a"  and  "b"  be positive real numbers   a >= 0,   b >= 0.   Prove that   a^4 + b^4 >= a^3*b + a*b^3.

        Problem 3.  Prove that if  |x+3| < 1/2 ,   then  |4x+13| < 3.


Math circle level problems on inequalities

        Problem 1.  If   3(a^2 + b^2 + c^2) = (a+b+c)^2,   find the relation between  a,  b  and  c.

        Problem 2.  If   a + b + c = 3   and   a%5E2+%2B+b%5E2+%2B+c%5E2 =3,   find   a%5E2022+%2Bb%5E2022+%2Bc%5E2022.

        Problem 3.  Let  x, y, z  be three positive real numbers such that   x^2 + y^2 + z^2 = 2(xy + xz + yz).
                           Prove that   x + y + z + 1/(xyz) > 4.


Math Olympiad level problems on inequalities

        Problem 1.  One day Vani caught fishes of weight  100 kg.  The total weight  3  largest fishes is  35 kg,
                           and total weight of  3  smallest fishes is  25 kg.  How many fishes Vani caught in total ?

        Problem 2.  Let  A = {n ∈ Z | |n| ≤ 24}.  In how many ways can two distinct numbers  x  and  y  be chosen  (simultaneously)  from  A
                           such that their product is less than their sum?
                           In more precise fashion,  what is the number of such ordered pairs  (x,y)  do exist,
                           where integer numbers  x,  y  are from set  A  and are distinct.

        Problem 3.  Charlie has a collection of books that he wishes to display in a narrow bookcase with shelves of width  56 cm.
                           The thickest books are no more than  16 cm wide and,  when placed side by side,  the entire collection takes up  2.4 m.
                           Find the minimum number of shelves required to guarantee that all of the books can be displayed in the bookcase.


Entertainment problems on inequalities

        Problem 1.  If a triangle has the perimeter of  78 cm and the longest side is  26 cm,
                           then the triangle is equilateral.

        Problem 2.  If a pentagon has the perimeter of  100 cm and the longest side is  20 cm,
                           then all the sides of this pentagon have the same length.

        Problem 3.  The entire freshman class of  Wannago  High  School is going on a field trip.
                           The number of buses needed is a function of the number of people going.
                           The school bus company has buses that can hold  36 passengers,  and there will be two adults per bus.
                           How many buses will be needed for  232 students?

        Problem 4.  I own a large truck,  and my neighbor owns four small trucks that are all identical.
                           My truck can carry a load of at least  400  pounds more than each of her trucks,
                           but no more than  7/15  of the total load her four trucks combined can carry.
                           Based on these facts,  what is the greatest load I can be sure that my large truck can carry,  in pounds?


Below is the list of lessons on domains of functions with solved problems


Domain of a function which is a quadratic polynomial under the square root operator

1)  y = sqrt%28x%5E2-4%29;  2)  y = sqrt%28-x%5E2%2B4%29;  3)  y = sqrt%28%28x-1%29%2A%28x-2%29%29;  4)  y = sqrt%28-%28x-2%29%2A%28x-3%29%29;  5)  y = sqrt%28x%5E2+-+4x+%2B+3%29;  6)  y = sqrt%28-x%5E2+%2B+2x+%2B+3%29;  7)  y = sqrt%28x%5E2+%2B+1%29;  8)  y = sqrt%28x%2F%28x%5E2+-+1%29%29.


Domain of a function which is a high degree polynomial under the square root operator

    1)  sqrt%28%28x-1%29%2A%28x-2%29%2A%28x-3%29%29     2)  sqrt%28%28x-1%29%2A%28x-2%29%2A%28x-3%29%2A%28x-4%29%29.


Domain of a function which is the square root of a rational function

    1)  y = sqrt%281%2F%28x-1%29%29;   2)  y = sqrt%28%28x-1%29%2F%28x-2%29%29;   3)  y = sqrt%28-%28%28x-1%29%2F%28x-2%29%29%29;   4)  y = sqrt%28%28x-2%29%2F%28%28x-1%29%2A%28x-3%29%29%29;   5)  y = sqrt%28%28%28x-1%29%2A%28x-3%29%29%2F%28%28x%2B1%29%2A%28x-2%29%29%29;   6)  y = sqrt%28%28x%5E2-2%29%2F%28x%5E2-4%29%29.

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