Chapter 6 Solving and Graphing Linear Inequalities By Michelle Lagana

What is an inequality? It contains a: > (greater than) < (less than) > (greater than or equal to) < (lesser than or equal to) instead of an = (equal sign)

So what does it look like graphically? X<4 (means that X represents ALL values LESS than 4) X<4 (means that X represents ALL values LESS than and INCLUDING 4) *** ***Remember that when graphs INCLUDE the value, there is a closed circle

Solving Inequalities Inequalities are just like normal equations but they have a few extra rules… So when solving them… 1. write the equation 2. ISOLATE the variable like you would for normal algebra problems 3. BEWARE of dividing and multiplying by negatives (and remember to just FLIP the sign)

Practice Time! Solve: x + 5 > 3 Subtract 5 from both sides to isolate the x x > 3 – 5 This becomes x > -2

More Practice! Ex. -(1/6)b < 3 1. Divide both sides by –(1/6) [-(1/6)b] / (-1/6) < 3 / (-1/6) Remember, whenever you divide or multiply a negative SWITCH the sign b > -18

Compound Inequalities A compound inequality consists of TWO inequalities connected by the word AND (if they meet) or OR (if they point in opposite directions) Compound Inequalities basically describe a range with a minimum and a maximum (as it is where the two inequalities OVERLAP), where as normal inequalities just describe either a minimum or a maximum

Solving compound inequalities Ex. (compound inequality involving AND) -3 < 2x + 1 < 5 Remember: you want to isolate the X In order to do this, whatever you do to one side, you must to do the other Subtract 1 from both sides < 2x < 5 -1 Divide each side by 2 -4/2 < 2x /2 < 4 /2 So: -2 < x < 2

Solving compound inequalities Ex. (compound inequality involving OR) x – 1 8 Solve each one just as you would a normal inequality x – – 3 So: x 5

Absolute-Value Equations Absolute-value equations are ones which are basically composed of two linear equations ex. l x + 2 l = 8 This actually equals: x + 2 = 8 or x + 2 = -8 x = 8 – 2 or x + 2 – 2 = -8 – 2 x = 6 or x = -10

Solving Absolute-Value Inequalities Ex. l x + 1 l – 3 > 2 l x + 1 l – > l x + 1 l > 5 (Now set up two linear equations) x + 1 > 5 or x + 1 < -5 (remember to flip the inequality sign when you change 5 to -5) x + 1 – 1 > 5 -1 or x < x > 4 or x < -6

Graphing Linear Inequalities in Two Variables Ex. 4x + 3y < 24 (isolate the y ) 4x + 3y – 4x < 24 – 4x 3y < 24 – 4x 3y /3 < (24 – 4x) /3 y < 8 – (4/3)x Graph y = 8 – (4/3)x Now shade everything to the left since y is LESS than 8 – (4/3)x