UC MATH 54 - Chapter 4 The Basics of Algebra

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Chapter 4 The Basics of Algebra ALGEBRA If you re planning to attend graduate school you ve probably had some sort of algebraic training in the dark reaches of your past Algebra is the fine art of determining how variable quantities relate to each other within complex functions and it dates back more than 3 000 years to ancient Babylon If it seems like it s been 3 000 years since you last studied algebra or even if you flunked algebra last year fear not This chapter is devoted to reintroducing you to the basic algebraic operations that you will need to execute on test day In the next chapter we will outline for you ways to subvert the rules and solve algebraic problems with strategies designed to reduce the amount of algebra you have to do But it is impossible and not in your best interests to ignore algebra altogether And even on problems that algebra can be subverted many times these problems can be made easier to work with after applying some basic algebra work Know the Lingo Any letter in an algebraic term or equation is called a variable you don t know what its numerical value is It varies Until you solve an equation a variable is an unknown quantity Any number that s directly in front of a variable is called a coefficient and the coefficient is a constant multiplied by that variable For example 3x means three times x whatever x is Combining Like Terms If two terms have the same variables or series of variables in them they re referred to as like terms You can combine them like this 6a 4a 10a 13x 7x 6x For example if you have six apples in one hand and four in the other you have a total of 10 apples and a pair of humongous hands Solving an Equation Whenever a variable appears in an equation and you have to find the value of the variable you have to isolate it To isolate a variable you must use mathematical operations to put all the terms that contain the variable on one side of the equals sign and all terms that don t contain the variable on the other Then you ll need to manipulate the side of the equation with the variable to find the value the question asks for In order to solve an equation you must rely on the following paramount rule of algebraic manipulation You can do anything you want to an equation as long as you do exactly the same thing to both sides Here s How to Crack It In order to get all variable terms on one side of the equals sign and all constant terms on the other add 5 to both sides The new equation is Now manipulate the new equation to isolate the variable by dividing both sides of the equation by 4 4x 24 x 6 4 6 5 19 24 5 19 19 19 Check Always Check After manipulating an equation using algebra on the GRE it always pays to check your work by replacing the variable with the value you found for the variable Let s do it Keep in mind that solutions to equations don t always have to be integers so don t be concerned if your result is a fraction As long as it works when you plug it back into the equation you re fine Here s How to Crack It Get all the variables onto the left side of the equation by adding 2m to both sides then move all the constants to the right side by subtracting 7 from both sides 4m 7 2m 16 2m 2m 6m 7 16 6m 7 7 16 7 6m 9 Now isolate the variable by dividing both sides of the equation by 6 Check your work just to make sure 6 7 16 3 13 13 Solving an Equation Quick Quiz Solve for x in each of the following equations 1 2 Explanations for Solving an Equation Quick Quiz Add 5 to both sides of the equation and then divide both sides of the equation by 2 to find that x 8 Subtract 3 from both sides and then divide both sides of the equation by 5 to find that x 2 3 Subtract 12x from both sides of the equation to find that 4 8x Divide both sides of the equation by 8 to find that x 4 5 Subtract x from both sides of the equation to find that 7x 9 2 Now add 9 to both sides of the equation to find that 7x 7 Divide both sides of the equation by 7 to find that x 1 Subtract 5 from both sides of the equation and then multiply both sides of the equation by 2 to find that x 28 Inequalities Inequality symbols are used to convey that one number is greater than or less than another The symbols used in inequalities are as follows means is greater than means is less than means is greater than or equal to means is less than or equal to Even though the two sides of an inequality aren t equal you can manipulate them in much the same way as you do the expressions in regular equations when you have to solve for a variable Here s How to Crack It Adding and subtracting take place as usual like this 5b 3 2b 9 3b 3 9 3b 12 At this point because the coefficient of b is positive divide both sides by 3 and get the final range of values for b To check your solution for inequality problems try a number that is greater than the value found for the variable In this case b 4 so replace the variable with a number such as 5 and check the solution b 4 5 5 3 2 5 9 25 3 10 9 22 19 Flip That Sign The only difference between solving equalities and inequalities is this one very important rule Whenever you multiply or divide both sides of an inequality by a negative number you must flip the inequality sign Try a problem 15 10 2 0 1 8 24 Here s How to Crack It Manipulate the problem as you would a regular equality by subtracting 5 from both sides of the equation to find that 5 11p 9 11p 4 To isolate the variable you must divide both sides of the inequality by 11 Because you are dividing by a negative number make sure to flip the sign Any value that is greater than is a possible value of p so you should check the box next to 0 and every other box with a number that s greater than zero See how important that little rule is If you didn t know about it you might have picked all the numbers that were less than which would be the exact opposite of the correct answers And that would have been unfortunate Range of Inequalities Some GRE questions present two inequalities and ask for the range of possible values when the inequalities are …


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