STUDY GUIDE Math 0 To the students: To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition When you study Algebra, the material is presented to you in a logical sequence. Many ideas are developed, left, and then returned to when your knowledge is broader. Many different kinds of problems have similar instructions. This presents great difficulty when trying to prepare for a final exam or keep up in the next Math class. You have mastered all the skills, but which one do you use in a specific problem? This guide was written to help you to re-organize your knowledge into a more usable form. When you are faced with a problem that begins, Solve for x. What should you do? As you will see, there are at least 9 different situations where you have seen that instruction. This guide will give you the key questions to ask yourself in order to decide what procedure to use. The main steps that are involved are included. The questions are asked in the ORDER that you should ask them. Each is referenced with a section number (or if only part of the section is involved, the specific page or problem number is given.) To use this guide effectively, you should first read through the guide. Each reference to a section should be examined carefully. Can you make up a problem like the one being described? Would you know how to solve that problem without any clues? Look at the problem or section referenced. Is it like yours? Can you work those easily? If so, go on to the next topic. If not, highlight that line with a marker for further study. Perhaps you should put an example problem on a 3 by 5 card (include the page number) for practice later. Now read the section again carefully. Work the examples and select a few similar problems from the exercises (odd ones so you can check the answers) for practice. When you finish a whole type (i.e. Solve for x) mix your 3 by 5 cards and treat them like a test. Simply verify that you know how to start the problem. Any that you miss will direct you back to the sections where you need further study. If you need further help, consider asking for a tutoring appointment in the Math Lab. When you know what SPECIFIC topics present a problem for you, you can make a tutoring session much more effective and be of help so your tutor can know what help you need. See the Math Lab Coordinator early in the semester to fill out an application for tutoring.
SOLVE FOR X Is there more than one letter? - Treat all letters EXCEPT the one you are solving for as if they were numbers. Is there an absolute value symbol in the equation? - Use the definition to separate into cases. In the equation quadratic in form? - Rewrite using u (or other) letter for the variable expression that is to be the middle term then solve as an ordinary quadratic, then substitute back into the equation involving u and x to find x. Is there a variable under a radical? - Solve for the radical first, then square both sides of the equation and simplify. You MUST check the solutions because some may not check in the original. Are there one or more fractions in the equation? - Multiply both sides of the equation by the denominator (or LCD.) Always be sure that no denominator can be zero. 3 Is there an x or higher power of x? - The only way we could work this would be to gather all terms on one side of the equation and then factor. - Use the Zero Product Principle to set each factor equal to zero, and then solve. There might be as many solutions as the highest power of x. Is x the highest power of x? Use any of the following: 1) Try factoring, it sometimes works ) Complete the square. WARNING: If a perfect square equals a negative number, quit. There is no real solution. 3) Put the equation in standard form and apply the Quadratic Formula. a) Simplify the radical, if possible. b) If the radical contains a perfect square, rewrite the solution twice, once with a + and once with a -, then simplify further. c) If the radical contains a negative number there is no real solution. Are there only x terms and constants? 1) Remove parentheses using the Distributive Law and simplify both sides. If you now find an x term, see instruction above. ) Using the Addition Property of Equality, gather the x terms on one side of the equation. 3) Using the Addition Property of Equality, gather the constants on the other side. 4) Divide both sides of the equation by the coefficient of x. 5) If the x terms disappear and: a) You get nonsense such as 5 =, then there is NO SOLUTION. b) You get a true statement such as 6 = 6, then ANY NUMBER is a solution. (This is called an identity). Sec. 1.5 Sec. 4.3 Sec. 8.4 Sec. 7.6 Sec. 6.6 Sec. 5.7 Sec. 5.7 Sec 8.1 Sec. 8. Sec. 1.4
Is there a logarithmic expression? Sec. 9.5 Is this an inequality? Is there an x or higher power? Gather all terms on one side and factor. Do a SIGN ANALYSIS to find the solutions. Is there no higher power of x? (than the first power) Treat it as an equation except that when you multiply or divide by a negative you must REVERSE the inequality. Graph the solution. Is this a compound of inequality? (It has inequality symbols) Perform operations on all 3 parts to isolate the x value. Translate and and or into intersection and union of sets. Absolute value inequality. Translate into inequalities without absolute value sign and solve. Rational inequality, boundary points. Is there a variable in the numerator or denominator of a fraction? Find all boundary points and test a point in each interval. Sec. 8.5 Sec. 4.1 Sec. 4. Sec. 4.3 Sec. 8.5 To solve linear equations in unknowns there are (equally good) methods. Each eliminates one variable in the first step. 1) Substitution ) Addition Method (You can observe the approximate solution by graphing both equations on the same graph. The solution is the coordinates of the point where the lines intersect.) What can go wrong? a) You lose BOTH variables in the first step and end up with nonsense like 0 = 7. There is NO SOLUTION. (In this case the lines on the graph would be parallel, so they don t meet at all.) We call this an inconsistent system. b) You lose both variables in the first step and end up with truth like 0 = 0. The answer is that there are MANY SOLUTIONS. (In this case if you graphed the lines, one would be superimposed over the other.) Both equations describe the same line so any point on the line represents a solution. To solve equations in unknowns with squares or higher powers of one or both variables. Use addition or substitution whichever allows you to eliminate a variable in the first step. Then use substitution to find the other part of the ordered pair. To solve 3 or more equations in 3 or more unknowns: Form a matrix and use elementary row operations. Note carefully to find inconsistent or dependent systems. Sec. 3.1 Sec. 3.1 Sec. 3.1 Sec. 3.1 Sec. 3.1 Sec. 10.5 Sec. 3.3, 3.4
COMPUTE OR EVALUATE Order of Operation 1) Work from the innermost grouping out. a) The numerator and denominator of a fraction are each groupings. b) An absolute value symbol is a grouping c) If the fraction is a complex, find SOME part that can be simplified and start there. ) Within a group: a) Exponentiate first. (SEE Exponential Expressions for more detail.) i. Only the closest possible base is raised to the power. To raise a negative base or a fractional base to a power REQUIRES parentheses. ii. Any base to the zero power is 1. Any base to a negative iii. power indicates that you take the reciprocal. A fractional exponent can be broken into an integer power and a root. b) Multiply and divide, moving from left to right. c) Simplify signs, if necessary. d) Algebraic addition is last Sec. 1.1 Sec. 1. Sec. 1.6 Sec. 7. Problems using Logarithms 1) Write the expression in terms of the logs of single numbers. ) Write each number in scientific notation using base 10 for ordinary numbers. Use natural logs if the problem involves power of e. 3) Look up the log of each number using the table of logs in the book (Mantissa) and write the log of each power of 10 by inspection (Characteristic.) 4) Simplify the expression into one with a positive mantissa and a single characteristic. 5) Write the answer in scientific notation using the body of the table for the mantissa. A Logarithmic Expression Use the definition of logarithm to write in exponential form, and then fill in the missing number. Sec. 9.3
GRAPH To graph ANY equation involving x and y: 1) Make a table for x and y. ) Pick at least 5 values, some negative, for x. (Occasionally, it may be convenient to pick some values for y.) 3) Using the formula given to you, complete the table. (Substitute each value into the formula then compute the remaining value.) It is particularly useful to substitute 0 for x to find the y-intercept (s) and 0 for y to find the x- intercept (s). 4) Plot the points from your table on the graph. 5) Connect the points smoothly moving from left to right To graph an equation like x = 4 (or any number.) All x values are 4; pick anything at all for y. The result will be a vertical line. To graph an equation like y = 7 (or any number.) All y values are 7, pick anything at all for x. The result will be a horizontal line. To graph an INEQUALITY with absolute value on a number line (only one variable). Find critical values and test intervals. To graph an INEQUALITY: 1) Graph as above but dot in line or curve. ) Pick any point well away from the dotted edge. (If the origin qualifies, it is an easy choice.) 3) Substitute the coordinates of your point into the inequality. a) If the test point makes the inequality true, shade in that side of your graph. b) If the test point does NOT make the inequality true, shade in the other side. 4) If the inequality allows =, (either or ) fill in the edge of the graph border solidly. To graph a system of inequalities: 1) Graph the first inequality as above. ) Using a different color, graph the second inequality on the same graph. 3) The answer is the region that is shaded with BOTH colors. Sec. 1.3 Sec..3 Sec..3 Sec. 4.3 Sec. 4.4 Sec. 4.1 The x-intercept of (a line) or curve is where it crosses the x-axis. To find its value, substitute 0 for y and then solve for x. The y-intercept is where the line or curve crosses the y-axis. To find its value, substitute 0 for x and then solve for y. If an equation can be put into the form y = mx + b, then it is a straight line. Sec..3 If an equation involves the second power of x or y or both, it may be a conic section. Sec. 10.1-10.4
To find the distance between two points, ( x 1, 1) and ( The Pythagorean Theorem gives us this formula: y x, y ) d = ( x x ) + ( y y ) 1 1 The midpoint of the line segment between two points, ( x1, y 1) and ( x, y ) is x 1+ x 1, y + y Sec. 10.1 Sec. 10.1 SLOPE The slope of a line can be determined in two ways. 1) If you know the equation of the line, solve it for y. The slope is the coefficient of x. ) If you know the coordinates of two points, ( x 1, y1 ) and ( x, y ) use the formula: y y1 m = x x 1 Sec..3 Sec..3 Parallel lines have the same slope. Perpendicular lines have slopes with product 1 Sec..4 SIMPLIFY Fractions: If there are no variables, see compute. Sec. 1.1 1. To add or subtract: a) Find the Lowest Common Denominator. Sec. 6. b) Change each fraction to an equivalent fraction by multiplying numerator and denominator by the same value. c) Add the numerators and use the common denominator. If there is a in front of a fraction be sure to distribute it to EVERY TERM in the numerator.. To multiply, factor numerators and denominators reducing where possible. Sec. 6.1 Leave the answer in factored form unless it is part of a larger problem. (i.e. must be added to other terms.) 3. To divide, FIRST invert the divisor, and then proceed as in multiplication. Sec. 6.1 4. If there is a fraction within a numerator or denominator, either: Sec. 6.3 a) Multiply numerator and denominator of the largest fraction by the LCD for all fractions OR b) Treat numerator and denominator as a grouping and simplify, then divide as indicated by the larger fraction. 5. Remember, no denominator of any fraction may ever be zero. Sec. 6.1 6. Always reduce final answers where possible by dividing common factors from Sec. 6.1 numerator and denominator.
Radicals: No negative under even index radical (COMPLEX) Sec. 7.1 1. Is the expression under the radical a perfect square? cube? Simplify. Sec. 71. Is there a factor of the expression under the radical that is a perfect square, cube etc? Factor it out and simplify. Remember, the radical always has a positive value, so if the expression is a variable, when it is negative the value of the RADICAL is its opposite. 3. Is there a fraction under the radical? Simplify the expression into a single fraction and separate into two separate radicals. 4. Is there a product or quotient of radicals? Perform the operations. 5. Are two radicals in a sum alike (same index and radicand?) Add using the coefficients of the radicals. 6. Is there a radical in a denominator? Rationalize it: a) If it is a single radical, multiply numerator and denominator by that radical. b) If there is a sum or two terms where one or both are radicals multiply the numerator and denominator by the CONJUGATE of the denominator. Sec. 7.3 Sec. 7.4 Sec. 7.3 Sec. 7.4 Sec. 7.5 Sec. 7.5 Exponential expressions 1) FIRST, review the rules of exponents. ) You may apply any appropriate rule to the expression, but the following strategies may be useful: a) Are there powers of other expressions? Use the Power (of a Product) rule to remove parentheses. b) Are there powers of exponential expressions? Use Power (of a Power) rule where appropriate. c) Are there like bases in numerator or denominator? Use the product rule to simplify (add exponents.) d) Are there like bases in both numerator and denominator? Divide (by subtracting exponents.) e) Are there negative exponents? Use the negative exponent rule to write the reciprocal. f) Write as a single fraction. g) Are you finished? Each exponent should apply to a single base. Each base should appear only once. There should be no negative exponents. Powers of numbers should be calculated. The fraction should be in lowest terms Sec 1.6
Logarithmic Expressions First, review the properties of Logarithms. Apply the properties one at a time until the goal is achieved. Complex Numbers a) We define that i = 1 b) Complex numbers are written as a + bi, where a and b are real numbers. c) To remove a complex number from the denominator of an expression, multiply by its conjugate. d) For power of 1, substitute (-1) for i squared as many times as possible or substitute 1 for i to the fourth. Scientific Notation Place the decimal point after the first non-zero digit and multiply by the appropriate power of 10. Simplify by using exponent rules on powers of 10. Sec. 9.3, 9.4 Sec 7.7 Sec. 1.7 FACTOR To factor a number means to write it as a product of primes (numbers that cannot be factored further.) Begin with any product and then break each number down until none can be factored further. To factor a polynomial: 1) Is there a factor common to all terms? Factor out the greatest common factor (term.) ) Are there 4 terms? Try factoring by grouping. 3) Is there a common pattern? a) Is this a difference of squares? b) Is this a perfect square trinomial? c) Is this the sum or difference of two cubes? When all else fails on a trinomial: 4) Perform a structures search. (This is an organized version of the trial factors from the text.) a) List all the possible ways to factor the first (squared) term. These are the column headings. b) In each column, list all the possible arrangements of the factors for the last (constant) term. (These form the rows.) c) Test each entry in your table using FOIL to see if this makes the middle term possible. (If there are no candidates, report that it DOES NOT FACTOR.) d) If you have a candidate, insert signs to try to match original. i. If the last sign (constant) is negative, the signs are different. ii. If the last sign is positive, the two signs are alike, Use the sign of the middle term. iii. iv. If none of the above works, go on searching for new candidates. If you exhaust the list and none work, report that it DOES NOT FACTOR. e) Check your solution. Check to be sure that none of the factors can be factored further Sec. 5.6 Sec. 5.3 Sec. 5.3 Sec. 5.5 Sec. 5.4
WORD PROBLEMS How to Solve Word Problems in Algebra By Mildred Johnson is an excellent and inexpensive resource. It is available in the bookstore. 1) Read through the problem to determine type. ) Draw a picture, if possible. 3) Write Let x be 4) Pick out the basic unknown and finish the above sentence. 5) Write as many other quantities as possible in terms of x and label them. 6) Is there STILL another unknown? If so, write, Let y be and complete the sentence. Write all other quantities in terms of x and y. You may need one or more of the formulas below to complete this. Sec. 1.5, 3., 5.7, 6.7, 10.5 Note: Tables are useful in many of these problems. Make one like the models in the text where appropriate. 7) Write any formula(s) that apply to this type of problem. a) d = rt (distance, time and speed) b) In wind or stream, when moving with the current, the speed is the sum of the speed of the craft and the current. c) i = Pr (interest for 1 year) d) Concentration of a solution (% target) (amount mixture) = amount target ingredient. e) (cost per item) (number of items) = value f) (denomination of a bill) (# of bills) = value g) consecutive numbers x, x + 1, x +, etc. h) consecutive ODD or EVEN numbers (The value of the first determines which) n, n +, n + 4 etc. i) In age problems, when they say in 5 years, write each age + 5 j) Work rate problems convert the time to do a job into the work done per time period by taking the reciprocal. THESE quantities can be added or subtracted. k) Geometric formulas are found on the back cover of the text. Ask your instructor which you are responsible for knowing. l) Fulcrum: Use weight x distance for each force. m) Cost Analysis n) Direct and Inverse Variation. 8) Use the formula or the words from the problem to write an equation. 9) Solve the equation for x (or x and y.) 10) REREAD the question. Write all the quantities from the original problem using the value for x as a key. 11) Answer the question asked. 1) Check the answer with the problem s original words. Discard any answers that don t fit. Sec. 6.8
OTHER TOPICS Functions and composites Sec.1,., 9. Inverse of a function Sec. 9. Arithmetic Sequences Sec 11. Geometric Sequences Sec 11.3 The Binomial Theorem Sec 11.4 Intersection and union of intervals Sec 4.