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Published byMartha Nelson Modified over 4 years ago

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Math 021

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An equation is defined as two algebraic expressions separated by an = sign. The solution to an equation is a number that when substituted into the equation makes it a true statement. For example, 8 is a solution to the equation x 2 – 9x = -8 since when x = 8 the equation becomes: 8 2 – 9(8) = -8 64 – 72 = -8 -8 = -8 which is a true statement However, 2 is not a solution since: 2 2 – 9(2) = -8 4 – 18 = -8 -14 = -8 which is a false statement

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A linear equation in one variable is any equation which contains a single variable and that variable is raised to the first power. The general form of a linear equation in one variable is ax + b = c where a, b, and c are real numbers.

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Let a, b, c be real numbers. If a = b, then a + c = b + c. The addition property allows you to add or subtract any term from both sides of an equation and the equation will remain equal. Examples – Solve the following using the addition property: a. x + 3 = 7 + 8b. 5x = 16 + 4x c. 7x - 5 = 8x + 10d. 10x – 5x = 4x – 11 e. 2(x + 6) = x – 3 f. 3(4x – 11) = -11(3 – x)

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Let a, b, c be real numbers, If a = b, then a∙c = b∙c. The multiplication property allows you to multiply or divide any non-zero number to both sides of an equation and the equation will remain equal. Examples – Solve the following using the multiplication property: a. 7x = 35b. 5x + 6x = 39 + 5

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c. -16(1-x) – 14x = –10 d. 5x – 4 = 26 + 2x e. 8x – 5x + 3 = x – 7 + 10 f. -2(5x – 1) – x = -4(x – 3)

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Multiply by a LCD to eliminate any fractions or multiply by a power of 10 to eliminate decimals Use the distributive property if necessary Combine like terms on the same side of the equal sign Use the addition property to isolate the term containing the variable on one side of the equation and the real number to the other Solve for the variable by using the multiplication property

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Examples – Solve each of the following: a. b. c. d. =

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e. 0.5x – 0.3 = 1.1 + 0.3x f. 0.15(4 – x) = 0.13(2 – x)

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A contradiction is a statement in mathematics that when completely simplified is false. A linear equation that simplifies to a contradiction has no solution. An identity is a statement in mathematics that when completely simplified is always true. A linear equation that simplifies to an identity has an infinite number of solutions, or all real numbers.

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a. 5x – 6x – 3 = -(x + 3) b. 3x + 3 + 5 = 2x + 2 + x c. 9(x – 2) = 7(x – 10) + 2x d. 5(x – 4) + x = 6(x – 2) – 8

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An absolute value equation is any equation that contains one or more absolute values. To eliminate absolute values, use the definition that if |x| = c, then x = c or x = -c Examples – Solve each of the following: a. |x + 3| = 7b. |x + 3| – 4 = 7 c. |2x – 5| + 1 = 6d. 3|5 – x| – 1 = 8 e. |3x + 1| + 10 = 6f. |2x| = -15

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