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Multiplying and dividing using integers
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An integer is any positive or negative whole number or zero The ASVAB often requires you to work with integers such as 6x = 36 (Don t forget, 6x is the same thing as 6 x) In multiplication and division, if the two terms being operated on (on either side of the equal sign) are both positive numbers or both negative numbers, the answer is a positive number If one number is negative and the other is positive, the answer is negative So to solve this problem, 6x = 36, you need to isolate x, so perform an inverse operation (remember, the inverse operation of multiplication is division): 6x -6 = 36 6 x = 6 The answer is a negative number because the two terms, 36 and 6, have different signs In an algebra equation, if the same letter is used more than once, it stands for the same number 3x + 2x = 10, the first x will never be a different number from the second x In this case, x = 2 (both times)
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8: Mathematics Knowledge
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You can only combine like terms when operating on algebraic expressions: 3x + 3x = 6x, but 3x + 3y doesn t equal 6xy, nor does x2 + x3 = x5 (see the section Explaining exponents, later in this chapter to find out more about algebra involving exponents)
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Solving multistep equations
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Not all algebra problems have one-step solutions (That would be too easy, and you wouldn t sweat nearly as much) Solving algebra problems on the ASVAB often requires you to perform several steps An example of a multistep equation is when x shows up on both sides of the equal sign Then you have to get rid of x from one side of the equation by moving an x from one side to the other You do this by performing the inverse operation Suppose you want to solve this equation: 3x + 3 = 9 + x Follow these steps: 1 To remove the x from one side of the equation, perform the inverse operation: 3x + 3 x = 9 + x x This equation can also be stated as 3x + 3 1x = 9 + 0 2 Perform the subtraction operation 2x + 3 = 9 3 To finish solving the problem, subtract 3 from each side of the equation 2x + 3 3 = 9 3 2x = 6 4 Divide both sides of the equation by 2 2x 2 = 6 2 x=3 When you have a variable by itself, such as x, it s always equal to 1 that variable (or one of that variable), like 1x, even if the 1 isn t written out In fact, any number is equal to 1 times itself, so you could also say 2 = 2 1 Sometimes this comes in handy when you re solving those algebra problems
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Explaining exponents
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Exponents are an easy way to show that a number is to be multiplied by itself a certain number of times For example, 52 is the same as 5 5 y3 is the same as y y y The number or variable that s multiplied by itself is called the base, and the number or variable showing how many times it is to be multiplied by itself is called the exponent or power Here are important rules when working with exponents: Any base raised to the power of one equals itself Example: x1 = x Any base raised to the zero power (except 0) equals 1 Example: x0 = 1 To multiply terms with the same base, you add the exponents Example: x2 x3 = x5 To divide terms with the same base, you subtract the exponents Example x5 (divided by) x2 = x3
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