Equations and Inequalities Chapter 2 Equations and Inequalities

Chapter Sections 2.1 – Solving Linear Equations 2.2 – Problem Solving and Using Formulas 2.3 – Applications of Algebra 2.4 – Additional Application Problems 2.5 – Solving Linear Inequalities 2.6 – Solving Equations and Inequalities Containing Absolute Values Chapter 1 Outline

Solving Linear Equations § 2.1 Solving Linear Equations

Properties of Equality For all real numbers a, b, ,and c: a = a Reflexive property If a = b, then b = a Symmetric property If a = b, and b = c, then a = c Transitive property

Combine Like Terms Like terms are terms that have the same variables with the same exponents. Like Terms -3x, 8x, - x 6w2, -12w2, w2 Unlike Terms 20x, x2, x3 6xy, 2xyz, w2

Combining Like Terms Example: 5a + 7a = 12a Determine which terms are like terms. Add or subtract the coefficients of the like terms. Multiply the number found in step 2 by the common variable(s). Example: 5a + 7a = 12a

Distributive Property For any real numbers a, b, and c, a(b + c) = ab + bc Example: 3(x + 5) = 3x + 15 (This is not equal to 18x! These are not like terms.)

Simplifying an Expression Use the distributive property to remove any parentheses. Combine like terms. Example: Simplify 3(x + y) + 2y = 3x + 3y + 2y (Distributive Property) = 3x + 5y (Combine Like Terms) (Remember that 3x + 5y cannot be combined because they are not like terms.)

Solve Linear Equations A linear equation in one variable is an equation that can be written in the form ax + b = c where a, b, and c are real numbers and a  0. The solution to an equation is the number that when substituted for the variable makes the equation a true statement.

Addition Property of Equality If a = b, then a + c = b + c for any real numbers a, b, and c. Example: Solve the equation x – 4 = -10. x – 4 = -10 x – 4 + 4 = -10 + 4 (Add 4 to both sides.) x = -6 Check: (-6) – 4 = -10

Multiplication Property of Equality If a = b, then a · c = b · c for any real numbers a, b, and c. Example: Solve the equation 12y = 15. ·12y = 15 · (Multiply both sides by ) 4 5 1

Multiplication Property of Equality Example: Solve the equation . (Multiply both sides by -4) (Simplify) x = -3 Check:

Solve Linear Equations Clear fractions. If the equation contains fractions, eliminate the fractions by multiplying both sides of the equation by the least common denominator. Simplify each side separately. Simplify each side of the equation as much as possible. Use the distributive property to clear parentheses and combine like terms as needed. Isolate the variable term on one side. Use the addition property to get all terms with the variable on one side of the equation and all constant terms on the other side. It may be necessary to use the addition property a number of times to accomplish this.

Solve Linear Equations Solve for the variable. Use the multiplication property to get the variable (with a coefficient of 1) on one side. Check. Check by substituting the value obtained in step 4 back into the original equation.

Solving Equations Example: Solve the equation 2x + 9 = 14. Don’t forget to check!

Solve Equations Containing Fractions Example: Solve the equation The least common denominator is 3.

Identify Conditional Equations, Contradictions, and Identities Conditional Equations: Equations that true for only specific values of the variable. Contradictions: Equations that are never true and have no solution. Identities: Equations that are always true and have an infinite number of solutions.

Solving Equations with Decimals Example: Determine whether the equation 5(a - 3) – 3(a – 6) = 2(a + 1) + 1 is a conditional equation, a contradiction, or an identity. Since we obtain the same expression on both sides of the equation, it is an identity.