University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

368

Pages:

18
Author:

inxj87o

Chapter 9 Equations, Inequalities and Problem Solving Part 1 Chapter Sections 9.1 – The Addition Property of Equality 9.2 – The Multiplication Property of Equality 9.3 – Further Solving Linear Equations 9.4 – Introduction to Problem Solving 9.5 – Formulas and Problem Solving 9.6 – Percent and Mixture Problem Solving 9.7 – Solving Linear Inequalities 2 § 9.1 The Addition Property of Equality Linear Equations Linear equation in one variable can be written in the form ax + b = c, a ≠ 0 Equivalent equations are equations with the same solutions in the form of variable = number, or number = variable 4 Addition Property of Equality Addition Property of Equality a = b and a + c = b + c are equivalent equations Example a.) 8 + z = – 8 8 + (– 8) + z = – 8 + – 8 z = – 16 (Add –8 to each side) (Simplify both sides) 5 Solving Equations Example 4p – 11 – p = 2 + 2p – 20 3p – 11 = 2p – 18 (Simplify both sides) 3p + (– 2p) – 11 = 2p + (– 2p) – 18 p – 11 = – 18 (Simplify both sides) p – 11 + 11 = – 18 + 11 p=–7 (Add –2p to both sides) (Add 11 to both sides) (Simplify both sides) 6 Solving Equations Example 5(3 + z) – (8z + 9) = – 4z 15 + 5z – 8z – 9 = – 4z 6 – 3z = – 4z (Use distributive property) (Simplify left side) 6 – 3z + 4z = – 4z + 4z (Add 4z to both sides) 6+z=0 6 + (– 6) + z = 0 +( – 6) z=–6 (Simplify both sides) (Add –6 to both sides) (Simplify both sides) 7 § 9.2 The Multiplication Property of Equality Multiplication Property of Equality Multiplication property of equality a = b and ac = bc are equivalent equations Example –y=8 (– 1)(– y) = 8(– 1) y=–8 (Multiply both sides by –1) (Simplify both sides) 9 Solving Equations Example 1 5 x= 7 9 1  5 7 x  =  7 7  9 35 x= 9 (Multiply both sides by 7) (Simplify both sides) 10 Solving Equations Example 8 x=6 3 83 8   x = 6  38 3 x = 16 (Multiply both sides by fraction) (Simplify both sides) 11 Solving Equations Recall that multiplying by a number is equivalent to dividing by its reciprocal Example 3z – 1 = 26 3z – 1 + 1 = 26 + 1 (Add 1 to both sides) 3z = 27 3 z 27 = 3 3 (Simplify both sides) z=9 (Simplify both sides) (Divide both sides by 3) 12 Solving Equations Example 12x + 30 + 8x – 6 = 10 20x + 24 = 10 20x + 24 + (– 24) = 10 + (– 24) (Simplify left side) (Add –24 to both sides) 20x = – 14 (Simplify both sides) 20 x − 14 = 20 20 (Divide both sides by 20) −7 x= 10 (Simplify both sides) 13 § 9.3 Further Solving Linear Equations Solving Linear Equations Solving linear equations in one variable 1) 2) 3) 4) Multiply to clear fractions Use distributive property Simplify each side of equation Get all variable terms on one side and number terms on the other side of equation (addition property of equality) 5) Get variable alone (multiplication property of equality) 6) Check solution by substituting into original problem 15 Solving Linear Equations Example 3( y + 3) = 2y + 6 5 5 ⋅ 3( y + 3) = 5(2 y + 6 ) 5 3 y + 9 = 10 y + 30 (Multiply both sides by 5) (Simplify) 3 y + (−3 y ) + 9 = 10 y + (−3 y ) + 30 (Add –3y to both sides) 9 + (−30) = 7 y + 30 + (−30) (Simplify; add –30 to both sides) − 21 7 y = 7 7 −3 = y (Simplify; divide both sides by 7) (Simplify both sides) 16 Solving Linear Equations Example 5x – 5 = 2(x + 1) + 3x – 7 5x – 5 = 2x + 2 + 3x – 7 (Use distributive property) 5x – 5 = 5x – 5 (Simplify the right side) Both sides of the equation are identical. Since this equation will be true for every x that is substituted into the equation, the solution is “all real numbers.” 17 Solving Linear Equations Example 3x – 7 = 3(x + 1) 3x – 7 = 3x + 3 (Use distributive property) 3x + (– 3x) – 7 = 3x + (– 3x) + 3 (Add –3x to both sides) –7=3 (Simplify both sides) Since no value for the variable x can be substituted into this equation that will make this a true statement, there is “no solution.” 18