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Solving Equations and Inequalities Vocabulary An equation is a sentence stating two quantities are equal. An algebraic equation is an equation that includes one or more variables. An equivalent equations is an equation that have the same solution(s). For example: x + 2 = 6, if x = 4 both sides of the equation are equal. Isolate – to solve an equation you must isolate the variable (i.e. get the variable alone to one side of the equation). An inverse operation undoes another operation by performing the opposite operation (i.e subtraction is the inverse operation of addition, multiplication is the inverse operation of division). 2 Properties of Equality – producing equivalent equations Addition Property of Equality: Adding the same number to each side of an equation produces an equivalent equation. x–3=2 x–3+3=2+3 Subtraction Property of Equality: Subtracting the same number to each side of an equation produces an equivalent equation. x+3=2 x+3-3=2-3 3 Properties of Equality – producing equivalent equations Multiplication Property of Equality: Adding the same number to each side of an equation produces an equivalent equation. 𝑥 3 𝑥 3 =2 ∙3=2∙3 Division Property of Equality: Dividing the same number to each side of an equation produces an equivalent equation. 5x = 20 5𝑥 𝟓 = 20 𝟓 4 Solving One-Step Equations 5 Solving an One-Step Equation Solving an equation with subtraction Solving an equation with addition x + 13 = 27 -7 = b - 3 isolate the variable -7 + 3 = b – 3 + 3 undo subtraction by adding the same number from each side Addition Property of Equality -4 = b Simplify each side of the equation isolate the variable x + 13 – 13 = 27 – 13 undo addition by subtracting the same number from each side Subtraction Property of Equality x = 14 Simplify each side of the equation Check it! x + 13 = 27 14 + 13 = 27 27 = 27 Check it! -7 = b - 3 Substitute the answer into original equation to check it. -7 = -4 - 3 -7 = -7 Substitute the answer into original equation to check it. 6 Solving an One-Step Equation Solving an equation with multiplication 4x = 28 isolate the variable 4𝑥 4 undo multiplication by dividing the same number from each side Division Property of Equality = 28 4 x=7 Simplify each side of the equation Check it! 4x = 28 4 ∙ 7 = 28 28 = 28 Substitute the answer into original equation to check it. Solving an equation with division 𝒙 𝟒 𝒙 𝟒 = -9 isolate the variable ∙ 𝟒 = -9 ∙ 4 undo division by multiplying the same number from each side Multiplication Property of Equality x = -36 Check it! 𝒙 = -9 𝟒 −𝟑𝟔 = -9 𝟒 -9 = -9 Simplify each side of the equation Substitute the answer into original equation to check it. 7 Solving an One-Step Equation solve equations with rational coefficients Solving an equation using reciprocal aka multiplicative inverses Two numbers with a product of 1 are called multiplicative inverses or reciprocals. Inverse Property of Multiplication The product of a number and its multiplicative inverse is 1. 3 4 𝑎 𝑏 Example: × = 1 ∙ = 1, a,b ≠ 0 4 𝟒 𝒎 𝟓 3 𝑏 𝑎 isolate the variable = 28 𝟓 𝟒 5 𝒎 = (28) 𝟒 𝟓 4 m = 35 Check it! Multiply each side by 5/4, the reciprocal of 4/5 Inverse Property of Multiplication Simplify each side of the equation 8 Solving Two and Multi -Step Equations 9 Understanding Two-Step To solve two-step equations, you use the properties of equality and inverse operations to form a series of simpler equivalent equations. You can use the properties of equality repeatedly to isolate the variable. Multi-Step To solve two-step equations, you use the properties of equality, inverse operations, and properties of real numbers to form a series of simpler equivalent equations. You use the properties until you isolate the variable. 10 You can undo the operations in the reverse order of the order of operations. 2x + 3 = 15 2x + 3 – 3 = 15 – 3 subtract 3 from each side 2x = 12 simplify 2𝑥 2 Divide each side by 2 = 12 2 x=6 Simplify Check 2x + 3 = 15 2(6) + 3 = 15 15 = 15 Solving Two-Step Equations Step 1: Undo the addition or subtraction first Step 2: Then undo the multiplication or division 11 Solving Equations with Variables on Both Sides 12 How To Get Started There are variables terms on both sides of the equation. Decide which variable term to add or subtract to get the variable on one side only. To solve equations with variables on both sides, you can use the properties of equality and inverse operations to write a series of simpler equivalent equations. 13 Solving an Equation with Variables on Both Sides 5x = 2x + 15 Original equation 5x – 2x = 2x +15 – 2x Subtract 2x from each side Subtraction Property of Equality 3x = 15 Simplify 𝟑𝒙 𝟏𝟓 = 𝟑 𝟑 Divide each side by 3 Division Property of Equality x=5 Simplify 14 Solving Inequalities 15 What is an inequality An inequality is a mathematical sentence that compares quantities Words Symbols x is less than two x<2 x is greater than or equal to four x≥4 Inequalities Words Symbols • is less than • is greater than • is less than or equal to • is greater than or equal to • Is fewer than • is more than • is no more than • is no less than • exceeds • is at most • is at least < > ≤ ≥ 16 Addition and Subtraction Properties of Inequality Addition and Subtraction Properties of Inequality When you add or subtract the same number from each side of an inequality, the inequality remains the same. For all numbers a, b, and c, 1. If a > b, then a + c > b + c and a – c > b – c 2. If a < b, then a + c < b + c and a – c < b – c Example 2<4 6>3 2+3<4+3 6–4>3–4 5<7 2>1 17 Solve Inequalities using Addition and Subtraction x + 3 > 10 Original equation -6 ≥ n – 5 Original equation x + 3 – 3 > 10 – 3 Subtract 3 from each side -6 + 5 ≥ n – 5 + 5 Add 5 to both sides x>7 Simplify -1 ≥ n Simplify Therefore, the solution is x > 7 Therefore, the solution is -1 ≥ n or n ≤ -1 18 Multiplication and Division Properties of Inequality, Positive Numbers Multiplication and Division Properties of Inequality, Positive Number When you multiply or divide each side of an inequality by a positive number, the inequality remains the same. For all numbers a, b, and c, where c > 0 𝑎 𝑏 > 𝑐 𝑐 𝑎 𝑏 1. If a < b, then ac < bc and < 𝑐 𝑐 1. If a > b, then ac > bc and 19 Solve Inequalities using Multiplication and Division, positive numbers 8x ≤ 40 Original equation 𝒅 𝟐 8𝑥 40 ≤ 8 8 Divide each side by 8 2 x≤5 Simplify d > 14 The solution is x ≤ 5. You can check this solution by substituting 5 or a number less than 5 into the inequality Original equation >7 𝑑 2 > 2(7) Multiply both sides by 2 The solution is d > 14. You can check this solution by substituting a number greater than 14 into the inequality 20 Multiplication and Division Properties of Inequality, Negative Numbers Multiplication and Division Properties of Inequality, Negative Number When you multiply or divide each side of an inequality by a negative number, the inequality symbol must be reversed for the inequality to remains true. For all numbers a, b, and c, where c < 0 𝑎 𝑐 𝑎 1. If a < b, then ac > bc and 𝑐 7>1 -2(7) < -2(1) REVERSE THE SYMBOLS -14 < -2 1. If a > b, then ac < bc and Example < > 𝑏 𝑐 𝑏 𝑐 -4 < 16 −4 16 > −4 −4 1 > -4 21 Solve Inequalities using Multiplication and Division, negative numbers 𝒙 ≤𝟒 −𝟑 −3 𝑥 −3 x ≥ -12 ≥ -3(4) Original equation Multiply each side by -3 and reverse the symbol Simplify The solution is x ≥ -12. You can check this solution by substituting -12 or a number greater than -12 into the inequality -6a ≥ -78 −6𝑎 −78 ≤ −6 −6 Original equation Divide both sides by -6 and reverse the symbol a ≤ 13 The solution is a ≤ 13. You can check this solution by substituting 13 or a number less than 13 into the inequality 22 Check Your UNDERSTANDING Solve each equation. Check your answer. Solving one-step equations Solving one-step equations 1. n + 6 = 8 6. 6c = 18 2. -2 = a + 6 7. -8x = 24 3. X - 5 = 6 8. 𝑝 9 9. 𝑎 12 4. -1 = c – 6 5. John F. Kennedy was the youngest President to be inaugurated. He was 43 years old. This was 26 years younger than the oldest president to be inaugurated – Ronald Reagan. Write and solve an equation to find out how old Reagan was when he was inaugurated. =9 = −3 10. A shark can swim at an average speed of 25 miles per hour. At this rate, how far 𝑑 can a shark swim in 2.4 hours? Use r = 𝑡 23 Check your UNDERSTANDING Solve each equation. Check your answer. Solving equations with Rational Coefficients and Multi-step equation Find the multiplicative inverse of each number. 11. 8 5 12. 9 Solve each equation. Check your solution 13. 1.6k = 3.2 14. 3 𝑎 8 = 4 7 12 40 15. -6 = 𝑥 3 16. Dillon deposited of his paycheck into 4 the bank. The deposit slip shows how much he deposited. Write and solve an equation to find the amount of his paycheck. Dillon deposited $46.50. 17. 3x + 1 = 7 18. -3y – 5 = 10 19. Syreeta wants to buy some CDs, each costing $14, and a DVD that costs $23. She has $65 to spend. Write and solve an equation to find how many CDs she can buy. 24 Check your UNDERSTANDING Solve each equation. Check your answer Solving equations with variables on both sides Express each equation as another equivalent equation. Justify your answer. 20. 4x + 8 = 2x + 40 21. 9x – 2 = 34 + 3x Solve each equation. Check your answer 22. 𝑥 3 − 15 = 12 + 𝑥 23. -7 + x = -8 – x 24. 11 + 4x = 7 + 5x 25 Check your UNDERSTANDING Solve each inequality. Check your answer 25. c + 4 < 8 30. 5x > 15 26. 14 + t ≥ 5 31. 27. y – 9 < 11 32. -7y > 28 28. c + 4 ≥ 17 29. An elevator can hold 2,800 pounds or less. Write and solve an inequality that describes how much more weight the elevator can hold if it is currently holding 2,375 pounds. 33. 2 3 4 5 < 𝑦 𝑡 −4 < −11 34. A pool charges $4 each visit, or you can buy a membership. Write and solve an inequality to find how many times a person should use the pool so that a membership is less expensive than paying each time. Membership is 3 months for $100. 26