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MM204 Unit 6 Seminar Agenda Solving Equations with fractions and decimals Translating English to Algebraic Expressions Inequalities

Steps for Solving Linear Equations in One Variable: 1) If there are fractions present in the equation, we will first determine the lowest common denominator (LCD) of all the fractions. We then will multiply all terms by that LCD. This will clear out all of the fractions. 2) We will then perform any simplification necessary by first applying the distributive property (when appropriate) and then combining any like terms. 3) By using the properties of addition and subtraction, we will isolate the variable terms. 4) By using the properties of multiplication and division, we will isolate the variable itself. 5) Check the solution by substituting it back into the original equation to see if it results in a true statement when both sides of the equation are simplified.

Example Solve for x. (3/4)x - 2/3 = (7/12)x Our LCD is 12 as all of the denominators will divide evenly into 12 so we will multiply all terms on both sides of the equal sign by 12 to clear out the fractions per step 1. 12(3/4)x - (12)(2/3) = 12(7/12)x (36/4)x - (24/3) = (84/12)x Next just simplify each fraction by reducing to lowest terms. 9x - 8 = 7x Now this equation looks a lot more familiar to us and we can apply the additive inverse property to isolate the variable term. We will first add 8 to both sides of the equation. 9x = 7x + 8 9x = 7x + 8 Next we will subtract 7x from both sides so that the variable terms are on the left side and the constant is on the right side. 9x - 7x = 7x x 2x = 8 Divide both sides by 2 to isolate the variable. 2x/2 = 8/2 x = 4

Check the answer? (3/4)x - 2/3 = (7/12)x; for x = 4 (3/4)(4) - 2/3 = (7/12)(4) 12/4 - 2/3 = 28/ /3 = 7/3 9/3 - 2/3 = 7/3 7/3 = 7/3 This is a true statement so we may safely conclude that our solution is correct.

Now let’s try one that looks a little different from the examples we have worked previously. Example: Solve for x. (2 + 3x)/5 + (3 - x)/2 = 3/10 The LCD = 10 so we will multiply all terms by (2 + 3x)/5 + 10(3 - x)/2 = 10(3/10) How we will handle this one is to write the fractional part of each expression before the terms within the ( ) as this will help us to more easily reduce to lowest terms. (10/5)(2 + 3x) + (10/2)(3 - x) = 30/10 Now reduce each fraction to lowest terms. 2(2 + 3x) + 5(3 - x) = 3 Apply the distributive property. 2(2) + 2(3x) + 5(3) + 5(-x) = x x = 3 Combine like terms on the left side x = x - 19 = x = -16

check (2 + 3x)/5 + (3 - x)/2 = 3/10; for x = -16 (2 + 3(-16))/5 + (3 - (-16))/2 = 3/10 (2 - 48)/5 + (3 + 16)/2 = 3/10 -46/5 + 19/2 = 3/10 LCD = 10. (-46/5)(2/2) + (19/2)(5/5) = 3/10 -92/ /10 = 3/10 3/10 = 3/10 True statement so our solution is correct. Questions??

Solve for x; present your answer in decimal form 0.6(x + 0.1) = 2(0.4x - 0.2) Since there are decimals within the ( ) then I’m going to multiply each term by 100 rather than 10, since 0.6 * 0.1 = 0.06 and multiplying by 10 will still leave a decimal value in that term. 100(0.6)(x + 0.1) = 100(2)(0.4x - 0.2) 60(x + 0.1) = 200(0.4x - 0.2) 60(x) + (60)(0.1) = 200(0.4x) - 200(0.2) 60x + 6 = 80x x = 80x x = 80x x - 80x = 80x x -20x = -46 Recall that we always solve for the positive value of the variable and never the negative so we will divide both sides by -20 to isolate the variable. -20x/-20 = -46/-20 x = 46/20 x = 2.3

Check? 0.6(x + 0.1) = 2(0.4x - 0.2); for x = ( ) = 2((0.4)(2.3) - 0.2) 0.6 (2.4) = 2( ) 1.44 = 2(0.72) 1.44 = 1.44 This is a true statement so our answer is correct. Please be aware that if we had rounded at any point in the equation that we might not have obtained ‘exact’ quantities to compare.

Key Words in English that Translate to Addition in Algebra: More Than Sum Of Increased By Added To Greater Than Plus

Example: Write '10 more than a number' using 'x' to stand for the unknown amount x Example: Write ‘the sum of 5 and a number’ using ‘x’ to stand for the unknown amount. 5 + x

Key Words in English that Translate to Subtraction in Algebra: Decreased By Less Than Subtracted From Smaller Than Fewer Than Diminished By Minus Difference Between Reduced By

Example: Write '7 decreased by a number' using 'x' to stand for the unknown amount. 7 - x Had we written x - 7 then that would mean x decreased by 7. Note: Remember that order IS important in subtraction so read your phrase back to yourself to be sure you have written it correctly and that it makes sense.

Key Words in English that Translate to Multiplication in Algebra: Double, Triple, etc. Twice Product Of Times

Example: Write '11 times a number' using 'x' to stand for the unknown amount. 11x

Key Words in English that Translate to Division in Algebra: Divided by Quotient Fractional amount of a number

Example: Write ‘the quotient of a number and 4’ using ‘x’ to stand for the unknown amount. 4/x Note: Remember that order IS important in division so read your phrase back to yourself to be sure you have written it correctly and that it makes sense.

Try this one Example: five more than one-third of a number. We will let x stand for ‘a number’. 1/3 of a number means 1/3 * that number or (1/3)x ‘five more’ means we are adding or 5 + Five more than one-third of a number then is: 5 + (1/3)x

One more example Example: one-fifth of a number reduced by double the same number. Again, we will do this in little chunks. We will let x stand for ‘a number’ and ‘the same number’. One-fifth of a number means 1/5 * that number or (1/5)x ‘reduced’ is a key word meaning subtraction so this gives us (1/5)x - Double the same number means we will multiply ‘the same number’ by 2 which will be 2x. Now we put it all together: One-fifth of a number reduced by double the same number is: (1/5)x - 2x

Inequalities Inequalities: An inequality is a relationship between quantities that states one quantity is greater than or less than another quantity. Example: 5 < 9 is read 'five is less than nine'. Read inequalities from left to right. Example: 9 > 5 is read 'nine is greater than five'. Read inequalities from left to right.

Inequality Symbols ≠ means 'is not equal to' < means 'less than‘ ≤ means 'less than or equal to‘ > means 'greater than  means 'greater than or equal to'

Steps for Solving a Linear Inequality: 1) If there are fractions present in the inequality, we will first determine the lowest common denominator (LCD) of all the fractions. We then will multiply all terms by that LCD. This will clear out all of the fractions. 2) We will then perform any simplification necessary by first applying the distributive property (when appropriate) and then combining any like terms. 3) By using the properties of addition and subtraction, we will isolate the variable terms. 4) By using the properties of multiplication and division, we will isolate the variable itself. If both sides of the inequality are multiplied or divided by a negative term the direction of the inequality symbol is reversed.

Example: Solve for x. 3x + 10 < 10x - 4 3x < 10x x < 10x x - 10x < 10x x -7x < x/-7 > -14/-7 Reverse the inequality symbol as there is division on both sides by a negative term x > 2