Time to start another new section!!! Time to start another new section!!!

P3: Solving linearP3: Solving linearequations and linear equations and linear inequalitiesinequalities

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 EquationsA linear equation in x is one that can be writtenin 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 forwhich the equation is true.

So, how many solutions are there to a linearSo, how many solutions are there to a linear equation in one variable???equation in one variable???

Let’s practice… Solve for the unknown:

2 2 3 3 1 5 2x x x

5 2.52

x

5 2 28 4y y

Let’s practice… Solve for the unknown:

K.T.D.!!!K.T.D.!!!5 2 28 4y y

(multiply both sides of the(multiply both sides of theequation by the L.C.D.)equation by the L.C.D.)

88 88

6y

Let’s practice… Solve for the unknown and support with grapher:

Now, how do we get Now, how do we get graphical graphical support???support???

4 4 2 3z z 13

z

Definition: Linear Inequality in xA linear inequality in x is one that can be writtenin the form

ax + b < 0, ax + b < 0, ax + b > 0, or ax + b > 0

where a and b are real numbers with a = 0

A A solution of an inequality in xsolution of an inequality in x is a value of x is a value of xfor which the inequality is true.for which the inequality is true.

The set of all solutions of an inequality is theThe set of all solutions of an inequality is thesolution set solution set of the inequality.of the inequality.

Properties of InequalitiesLet u, v, w, and z be real numbers, variables, or algebraicexpressions, and c a real number.

1. Transitive

2. Addition

3. Multiplication

(the above properties are true for < as well – there aresimilar properties for > and >)

If u < v and v < w, then u < w

If u < v, then u + w < v + wIf u < v and w < z, then u + w < v + z

If u < v and c > 0, then uc < vcIf u < v and c < 0, then uc > vc

Guided Practice:Solve the inequality:

When solving inequalities, don’t

When solving inequalities, don’t

forget to switch the inequality

forget to switch the inequality

sign whenever you multiply or

sign whenever you multiply or

divide by a negative number!!!

divide by a negative number!!! 3 1 2 5 6x x

72

x

Guided Practice:Solve the inequality, write your answer in intervalnotation, and graph the solution set:

–2 0

1 13 2 4 3x x

2x ( 2, )

Remember the KTD

“trick”!! L

CD=12

Guided PracticeSolve the double inequality, write your answer ininterval notation, and graph the solution set:

0 5

(–7, 5]

–7

2 53 53x

7 5x

Whiteboard Practice:Solve the inequality, write your answer in intervalnotation, and graph the solution set:

, 8

–11/7 0117

3 5 2 12 3m m

117

m

––

Whiteboard Practice:Solve the inequality:

7 8 3 2 4 7 2w w w

192

w 19( , )2

Whiteboard Practice: Solve and support with grapher:

1 5 13 4 2c c

1212 1212

5 0.7147

c Graphical Support?Graphical Support?

Homework: p. 28-29 11-27 odd, 35-53 odd