Questions on Algebra: Inequalities, trichotomy answered by real tutors!

The expression (x-5)(x-4) is a quadratic with two zeros, 4 and 5.

So there are three open intervals to test on each side of and between
the two zeros to determine whether on them the expression is always 
greater than or always less than 0.



Choose easiest test point x=0 on 
(0-5)(0-4) = 20 Thus the expression is greater 
than 0 on this interval.  So we will use it with the zero 4

Choose easiest test point x=4.5 on 
(4.5-5)(4.5-4) = -0.25 Thus the expression is less 
than 0 on this interval.  Thus we will not use it.

Choose easiest test point x=6 on 
(6-5)(6-4) = 2 Thus the expression is greater 
than 0 on this interval.  So we will use it with the zero 5

Answer: The expression is greater than 0 on 






Edwin

So, they want you solve inequality  (x-5)*(x-4) >= 0.


In this problem the function is presented as the product of two linear binomials, (x-5) and (x-4).


In order for the product of these two binomials would be non-negative, the factors/binomials 
should be of the same sign or zero.



First factor,  (x-5) is negative or zero in the area x <= 5 and non-negative or zero in the area x >= 5.

Second factor, (x-4) is negative or zero in the area x <= 4 and non-negative or zero in the area x >= 4.



Thus we see that both factors are negative or zero in the domain  x <= 4,

             and both factors are non-negative in the domain  x >= 5.


Thus the solution set is the union of two semi-infinite intervals  x <= 4 and x >= 5.


At this point, the solution is fully complete.


ANSWER.  The solution is the set  x <= 4  or  x >= 5.

         i.e. the area out of the interval between the roots of the function (but the solution set includes the roots).


To make sure that the solution is correct, you may plot the function.  See this plot below.


    


               Plot y = (x-5)*(x-4)



The function is a parabola opened upward. Its branches are in the upper (non-negative) half-plane out the interval
on x-axis between x-intercepts, but including the x-intercepts.


The part of the parabola between the roots is in the lover (negative) half-plane.


So, this figure confirms the logical reasoning in algebraic solution.

               Pasta   Tofu
Protein         8g     16g
Carbohydrates  60g     40g
Vitamin C       2g      2g
Cholesterol    60mg    50mg

Let introduce variable 'x':

Let 'x'  be the number of days of the month James eats pasta for lunch;

then the number of days James eats tofu for lunch is (30-x)   //  assuming 30 days in a month.


So, the problem is to minimize 

    H = 60x + 50(30-x)    (1)


under restriction

     8x + 16(30-x) >= 200,    (2)   (protein restriction)

    60x + 40(30-x) >= 960,    (3)   (carbohydrate restriction)

     2x + 2(30-x)  >=  40     (4)   (vitamin C restriction)

     0 <= x <=  30.           (5)


Let' simplify expression (1) and inequalities (2) - (5).


In simplified form, we want to minimize

    H = 10x + 1500            (1')

under restrictions

    -8x +  480 >= 200,  which we transform to  8x <= 280,    which we transform  x <= 35;     (2')

    20x + 1200 >= 960,  which we transform to  20x >= -240,  which we transform  x >= -12;    (3')

            60 >=  40,                                                                        (4')

     0 <= x <= 30.             (5')


Looking at these transformed restrictions, we see that in the domain  0 <= x <= 30 they (the restrictions)
are held at any value of x.


So, if we want to minimize (1'), we should take x = 0.


The solution is that James should not eat pasta and should eat tofu every day for lunch.

Indeed, with the given input data, it is clear that James can eat either pasta or tofu every day

    (a)  consuming protein will be more than 200 grams in either case;

    (b)  consuming carbohydrate will be more that 960 grams in either case;

    (c)  consuming vitamin C will be more than 40 grams in either case.


So, in reality, these three restrictions are not the restrictions, at all.


Therefore, if James wants to minimize cholesterol, he should eat tofu every day,
since tofu provides lesser cholesterol.

Equation  x^2 + y^2 = 25  describes a circle of the radius of 5 units,
centered at the origin of the coordinate system.


Inequality  x^2 + y^2 <= 25  describes the interior of this circle,
including the circle itself.


Equation y = |x|  describes two rays on the plane.
One ray bisects first quadrant; the other ray bisects the second quadrant.

So, inequality y >= |x|  describes the set of points on the coordinate plane between these rays.


The system of inequalities

    x^2 + y^2 <= 25,

    y >= |x|


describes the intersection of these two sets.


In other words, it is the interior of the circle, which is concluded between the two rays.

All the boundaries are included to the set.