Transcript Document
Section 3.4: Solving Equations with Fractions Steps for Solving Linear Equations in One Variable: 1. Eliminate parentheses first. 2. Get rid of fractions (this week!). 3. Collect Like terms. 4. Use the addition and/or subtraction properties to get all the letters on one side and constants on the other. 5. Use the multiplication and/or division properties to get the letter alone. 6. Check it! Section 3.4 To get rid of all the fractions in an equation: Find the LCD of all the fractions. Multiply both sides of the equation by the LCD. All the denominators will cancel with the LCD! Example: 2 x 3 2 2 x (6) (6) 3 2 2 x (2) (3) 1 1 (2)(2) = (3)(x) 4 = 3x 4/3 = x Multiply both sides by 6. All the denoms cancel. Fractions in Equations Example 1 2x 4 x 1 5 3 15 (15) (3) 1 2x 4x 1 (15) (15) 5 3 15 1 2x 4x 1 (5) (1) 1 1 1 3(1 + 2x) + 5(4 – x) = 1 3 + 6x + 20 – 5x = 1 23 + x = 1 23 + x – 23 = 1 – 23 x = -22 Multiply both sides by 15. All the denoms cancel. Solve as usual. Use distributive prop. Subtract 23 from both sides to get x alone. Fractions in Equations Example x 5 x 3 6 2 4 x 5 x 3 (12) (12) (12) 6 2 4 x 5 x 3 (2) (6) (3) 1 1 1 2(x + 5) = 6x + (3)3 2x + 10 = 6x + 9 2x - 6x + 10 = 6x - 6x + 9 -4x + 10 = 9 -4x + 10 - 10 = 9 - 10 -4x = -1 4x 1 4 4 1 x 4 Multiply both sides by 12. All the denoms cancel. Fractions in Equations Example 3 3 1 (x 2) (x 1) 4 5 5 3 3 1 (20) (x 2) (20) (20) (x 1) 4 5 5 3 3 1 (5) (x 2) ( 4) ( 4) (x 1) 1 1 1 (5)(3)(x - 2) + (4)(3) = (4)(1)(x + 1) 15(x - 2) + 12 = 4(x + 1) 15x - 30 + 12 = 4x + 4 15x - 18 = 4x + 4 15x - 4x - 18 = 4x - 4x + 4 11x - 18 = 4 11x - 18 + 18 = 4 + 18 11x = 22 x=2 Multiply both sides by 20. All the denoms cancel. Decimals Example - 3.5x + 1.3 = - 2.7x + 1.5 We can use the multiplication property to get rid of all the decimals. Notice if I multiply both sides by 10 . . . 10(- 3.5x + 1.3) = 10(- 2.7x + 1.5) -35x + 13 = -27x + 15 All the decimals have disappeared! -35x + 13 + 27x = - 27x + 15 + 27x -8x + 13 = 15 Move the -27x to the left side by adding. -8x + 13 – 13 = 15 – 13 -8x = 2 Move 13 to the right by subtracting. (-8x/- 8) = (2/-8) x = -1/4, which is the same as -.25 Solve for x by dividing each side by -8. Section 3.5: Translating Addition Phrases: More than, Sum of, Increased by, Added to, Greater than, Plus. Multiplication Phrases: Double, Twice, Product , Of, Times. Subtraction Phrases: Decreased by, Less than, Subtracted from, Smaller than, Fewer than, Diminished by, Minus, Difference between. Division Phrases: Divided by, quotient, half, third, fourth, fifth, sixth, etc... Section 3.5: Example of Translating #24 on page 148: The number of pounds of fish caught by Captain Jack was 813 pounds more than the amount of fish caught by Captain Sally. The amount of fish caught by Captain Ben was 623 pounds less than the amount of fish caught by Captain Sally. Section 3.5: Example of Translating #31 on page 149: Kentucky has about half the land area of Minnesota. The land area of Maine is approximately two-fifths the land area of Minnesota. Describe the land area of each of these three states. Section 3.6: Solving Inequalities Eliminate parentheses first. Get rid of fractions. Collect Like terms. Use the addition and/or subtraction properties to get all the letters on one side and constants on the other. Use the multiplication and/or division properties to get the letter alone. If we multiply or divide by a negative, we must flip the inequality symbol. Check it! Inequality Example Example: Solve for x. 6x + 5 < 29 6x + 5 - 5 < 29 - 5 Subtract 5 from both sides to get x term alone. 6x < 24 6x/6 < 24/6 x<4 Divide both sides by 6 to get x alone. Inequality Example 7 - 2(x - 4) < 7(x - 3) 7 - 2x + 8 < 7x – 21 15 - 2x < 7x – 21 15 - 2x - 7x < 7x - 7x – 21 15 - 9x < -21 15 - 15 - 9x < -21 – 15 -9x < -36 -9x / -9 > -36 / -9 x >4 Use the distributive prop. Graphing Inequalities To graph the solution on a number line: If we have < or >, draw a circle on the number in the solution. Shade left or right as the solution indicates. If we have or , draw a dot on the number in the solution and shade as the solution indicates. Graphing Inequalities Examples Thanks for Participating! AIM: tamitacker Read, read, read! Email me if you have questions.