Time to start another new section!!! P3: Solving linear equations and linear inequalities

Properties of Equality (let u, v, w, and z be real numbers, variables, or algebraic expressions) 1. Reflexive 2. Symmetric 3. Transitive 4. Addition 5. Multiplication u = u If u = v, then v = u If u = v, and v = w, then u = w If u = v and w = z, then u + w = v + z If u = v and w = z, then uw = vz Algebraic Properties

Solving Equations A linear equation in x is one that can be written in the form ax + b = 0 where a and b are real numbers with a = 0 A solution of an equation in x is a value of x for which the equation is true.  S S S So, how many solutions are there to a linear equation in one variable???

Let’s practice… Solve for the unknown:

K.T.D.!!! (multiply both sides of the equation by the L.C.D.) 88

Let’s practice… Solve for the unknown and support with grapher: Now, how do we get graphical support???

Definition: Linear Inequality in x A linear inequality in x is one that can be written in the form ax + b 0, or ax + b > 0 where a and b are real numbers with a = 0 A solution of an inequality in x is a value of x for which the inequality is true. The set of all solutions of an inequality is the solution set of the inequality.

Properties of Inequalities Let u, v, w, and z be real numbers, variables, or algebraic expressions, and c a real number. 1. Transitive 2. Addition 3. Multiplication (the above properties are true for < as well – there are similar properties for > and >) If u < v and v < w, then u < w If u < v, then u + w < v + w If u < v and w < z, then u + w < v + z If u < v and c > 0, then uc < vc If u < v and c < 0, then uc > vc

Guided Practice: Solve the inequality: W h e n s o l v i n g i n e q u a l i t i e s, d o n ’ t f o r g e t t o s w i t c h t h e i n e q u a l i t y s i g n w h e n e v e r y o u m u l t i p l y o r d i v i d e b y a n e g a t i v e n u m b e r ! ! !

Guided Practice: Solve the inequality, write your answer in interval notation, and graph the solution set: –20 Remember the KTD “trick”!! LCD=12

Guided Practice Solve the double inequality, write your answer in interval notation, and graph the solution set: 05 (–7, 5] –7

Whiteboard Practice: Solve the inequality, write your answer in interval notation, and graph the solution set:, 8 –11/ ––

Whiteboard Practice: Solve the inequality:

Whiteboard Practice: Solve and support with grapher: 1212 Graphical Support?

Homework:  p odd, odd