FOREWORD TO
Algebra PROBLEMS Algebra problems are most concerned with expressing word statements into one or more equations which may be solved as single-variable equations or systems of equations.
There are various types of problems, most of whom share similar solution processes.
This module is concerned with the solution of the problems with respect to theory. Quick solutions are not available for all questions. Join us for STEP 2 next semester for discussions on Calculator Techniques!
AGE PROBLEMS Assign the unknown as x and express the other variables in terms of x: Person 1 Person 2 (Person 3)
NOW x ax + b cx + d
PAST OR FUTURE AGE x ± no. of yrs ax + b ± no. of yrs cx + d ± no. of yrs
SUM
This leaves us with an equation with a single unknown: (x ± no. of yrs) + (ax + b ± no. of yrs) + (cx + d ± no. of yrs) = SUM
Other problems require the use of another variable, but most cases can be solved using a single variable.
AGE PROBLEMS See Pre-Test problem: John’s father is 5 times older than John and John is twice as old as his sister Alice. In two years time, the sum of their ages will be 58. How old is John now?
John Father Alice
NOW x 5x 0.5x
IN TWO YEARS x +2 5x + 2 0.5x + 2 58
(x + 2) + (5x + 2) + (0.5x + 2) = 58 Solving for x, x=8
WORK/rate PROBLEMS Determine the amount of job the person or unit can do by taking the reciprocal: TIME per unit job Unit 1 X hr/job Unit 2 Y hr/job (++ Units) Z hr/job
JOB per unit time X-1 job/hr Y-1 job/hr Z-1 job/hr
The total rate is the sum of the individual rates: đ?‘—đ?‘œđ?‘?
đ?‘—đ?‘œđ?‘?
đ?‘—đ?‘œđ?‘?
Total rate = X-1 â„Žđ?‘œđ?‘˘đ?‘&#x; + Y-1 â„Žđ?‘œđ?‘˘đ?‘&#x; + (Z-1 â„Žđ?‘œđ?‘˘đ?‘&#x;) Time for single job by all units = (Total rate)-1
WORK/rate PROBLEMS See Pre-Test problem: Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours to paint a similarly-sized house. How long would it take the two painters together to paint the house? Painter 1 Painter 2
TIME per unit job 12 hr/job 8 hr/job 1 đ?‘—đ?‘œđ?‘?
Total rate = 12 â„Žđ?‘&#x; +
1 đ?‘—đ?‘œđ?‘? 8 â„Žđ?‘&#x;
JOB per unit time 1/12 job/hr 1/8 job/hr
= 0.2083
đ?‘—đ?‘œđ?‘? â„Žđ?‘&#x;
Time for single job by all units = (0.2083 = 4.8 hours/job
FAST CALC (12-1 +8-1)-1 = 4.8
đ?‘—đ?‘œđ?‘? -1 ) â„Žđ?‘&#x;
integer PROBLEMS Assign the two variables as x and y, and formulate the appropriate expressions: Ax + By = M Cx + Dy = N These equations may be solved by Elimination, Substitution, or by Fast Calc thru MODE 5-1.
integer PROBLEMS See Pre-Test problem: Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers?
Let x be the smaller number Let y be the larger number 2y = 3 + 5x 4y + 3x =71 FAST CALC
Solving for x, y = {5,14}
ON FUNCTION
domain and range Determining the domain and range of a function involves determining the minimum, maximum, and null values of the function. Most simple function have for their domain the set of all real numbers. This is not the case, however, for radicals and rationals. For trigonometric functions, the domain and range depend on the type of function.
ON FUNCTION
domain and range For radicals: The domain is the set of all real numbers except the values of x that would make the radicand negative (and give a complex value for the radical). To find the domain: Set the radicand equal to zero and the roots of the expression is removed from the set of all real numbers as the domain of the function.
ON FUNCTION
domain and range See Pre-Test Problem: Find the domain of
2đ?‘Ľâˆ’8 . đ?‘Ľ 2 −16
Factoring and simplifying, 2đ?‘Ľâˆ’8 2(đ?‘Ľâˆ’4) 2 → → đ?‘Ľ 2 −16 (x + 4)(x – 4) (x + 4)
Finding the zeros of the denominator, (x + 4) = 0 → x = –4 The domain is therefore: R – {-4}
ON FUNCTION
domain and range For rationals: The domain is the set of all real numbers except the values of x that would make the denominator zero (which makes the function undefined).
To find the domain: Simplify the function by factoring, if possible. Find the range of x that makes the expression positive ( f(x) > 0).
ON FUNCTION
domain and range Sample problem: Find the domain of
1
.
đ?‘Ľ 2 −16
The denominator is positive for: đ?‘Ľ 2 − 16 > 0 đ?‘Ľ 2 > 16
x < -4 and x > 4 The domain is therefore: (-â&#x2C6;&#x17E;, -4) U (4, â&#x2C6;&#x17E;)
ON
vertical asymptotes Vertical asymptotes are values of the abscissa that the function can only approach and not reach. These are present in graphs of rational functions. These values of the abscissa are analogous to the roots of the denominator.
ON
horizontal asymptotes Horizontal asymptotes are not as strict as vertical asymptotes and the graph may intersect it at intermediate values. They graph usually only approaches the value as an asymptote at extreme values (as the function approaches negative or positive infinity).
ON
horizontal asymptotes Guidelines for horizontal asymptotes: 1) If the numerator and denominator have the same highest degrees, the horizontal asymptote is the ratio of the highest-degree coefficients of the numerator and denominator. 2) If the polynomial in the numerator has a lower degree that the denominator, the x-axis (y = 0) is the horizontal asymptote. 3) If the polynomial in the numerator has a higher degree than the denominator, there is no horizontal asymptote.
ON
sequences & series The general expression for sequences is:
an = a0 + (n â&#x20AC;&#x201C; 1)d
Most problems require the substitution of any number of these variables and the required value can be solved by a singular equation or a simple system of equations.
ON
sequences & series The general expression for sequences is:
an = a0 + (n â&#x20AC;&#x201C; 1)d
Most problems require the substitution of any number of these variables and the required value can be solved by a singular equation or a simple system of equations.
ON
sequences & series See Pre-Test problem: An arithmetic sequence has its 5th term equal to 22 and its 15th term equal to 62. Find its 100th term. an = a0 + (n – 1)d a5 = a0 + (5 – 1)d = 22 a15 = a0 + (15 – 1)d = 62 a0 + 4d = 22 a0 + 14d = 62 Solving simultaneously, a0 = 6 d = 4
ON
logarithms & exponentials Logarithms and exponentials can be converted interchangeably as demonstrated in the equations below: y = xn
logx y= n
Most problems can be solved by converting from one syntax to another, depending on the given conditions.
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sequences & series See Pre-Test problem: 3 Evaluate đ?&#x2018;Ľ = log 5 25. By direct input into calculator, 3 đ?&#x2018;Ľ = log 5 25 2
Solving for x â&#x2020;&#x2019; đ?&#x2018;Ľ = 3 , By principle, 3 đ?&#x2018;Ľ = log 5 25
5đ?&#x2018;Ľ =
3
25 =
3
đ?&#x;? đ?&#x2019;&#x2122; = đ?&#x;&#x2018;
52 =
2 53
NOTES ON
Algebra PROBLEMS There are more types of problems that can be explored that follow similar processes. Check out: - Money problems - Distance problems - Digit problems Watch out for our follow-up problems on College Algebra, and keep supporting our Quenched 2017, #Desire4Knowledge. Yours in academic empowerment, JPICHE-XU Department of Academics