3-30-16

< > < < Solving Inequalities < < < >

≥ : greater than or equal to An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

What do Inequalities mean? A mathematical sentence that uses one of the inequality symbols to state the relationship between two quantities.

Graphing Inequalities When we graph an inequality on a number line we use open and closed circles to represent the number. < < Plot an open circle ≤ ≥ Plot a closed circle

x < 5 means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5!

Numbers less than 5 are to the left of 5 on the number line. 5 10 15 -20 -15 -10 -5 -25 20 25 If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right. There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. The number 5 would not be a correct answer, though, because 5 is not less than 5.

Try to name ten numbers that are greater than or equal to x ≥ -2 means that whatever value x has, it must be greater than or equal to -2. Try to name ten numbers that are greater than or equal to -2

Numbers greater than -2 are to the right of -2 on the number line. 5 10 15 -20 -15 -10 -5 -25 20 25 -2 If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. The number -2 would also be a correct answer, because of the phrase, “or equal to”.

Solving an Inequality Follow the same rules and steps that we used to solve an equation. Always undo addition or subtraction first, then multiplication. Remember whatever is done to one side of the inequality must be done to the other side. The goal is to get the variable by itself.

Solve an Inequality w + 5 < 8 - 5 -5 w < 3 - 5 -5 w < 3 All numbers less than 3 are solutions to this problem! 5 10 15 -20 -15 -10 -5 -25 20 25

More Examples 8 + r ≥ -2 -8 -8 r -10 ≥ -8 -8 r -10 All numbers greater than-10 (including -10) ≥ 5 10 15 -20 -15 -10 -5 -25 20 25

More Examples 2x > -2 2 2 x > -1 2 2 x > -1 All numbers greater than -1 make this problem true! 5 10 15 -20 -15 -10 -5 -25 20 25

All numbers less than 8 (including 8) More Examples 2h + 8 ≤ 24 -8 -8 2h ≤ 16 2 2 h ≤ 8 All numbers less than 8 (including 8) 5 10 15 -20 -15 -10 -5 -25 20 25

Your Turn…. x + 3 > -4 x > -7 6d > 24 d > 4 2x - 8 < 14 Solve the inequality and graph the answer. x + 3 > -4 6d > 24 2x - 8 < 14 2c – 4 < 2 x > -7 d > 4 x < 11 c < 3

Very important…. <  > > < ≤  ≥ ≥  ≤ When you multiply or divide each side of an inequality by a negative number you always reverse or flip the inequality sign. <  > > < ≤  ≥ ≥  ≤

-7 > -4 7(-1) > 4(-1) RATIONALE -7 < -4 NOT TRUE! 7 > 4 You must change the inequality symbol -7 < -4

RATIONALE   NOT TRUE! You must change the inequality symbol -7 < -4

Solve the Inequality   Sign flipped because I divided by a negative number!! Sign flipped because I multiplied by a negative number!!

You Try!!!   x ≤ -2 x < -4 x < -5 x ≤ -64

3-31-16

Writing Compound Inequalities Write an inequality that represents the statement and graph the inequality. 1. x is greater than –4 and less than or equal to –2. l l l l l l l -6 -5 -4 -3 -2 -1 0

Writing Compound Inequalities Write an inequality that represents the statement and graph the inequality. 2. x is greater than 3 or less than –1. l l l l l l l -2 -1 0 1 2 3 4

Solving a Compound Inequality with And Solve the inequality and graph the solution. l l l l l l l l -3 -2 -1 0 1 2 3 4

Solving a Compound Inequality with And Solve the inequality and graph the solution. l l l l l l l l -5 -4 -3 -2 -1 0 1 2

Solving a Compound Inequality with And Solve the inequality and graph the solution. l l l l l l l l -4 -3 -2 -1 0 1 2 3

Solving a Compound Inequality with Or Solve the inequality and graph the solution. l l l l l l l l 3 4 5 6

Solving a Compound Inequality with Or Solve the inequality and graph the solution. l l l l l l l l -6 -5 -4 -3 -2 -1 0 1

Writing and Using a Linear Model 8. In 1985, a real estate property was sold for $172,000. The property was sold again in 1999 for $226,000. Write a compound inequality that represents the different values that the property was worth between 1995 and 1999.

4-1-16

Solving Absolute Value Equations

What is Absolute Value? The absolute value of a number is the number of units it is from zero on the number line. 5 and -5 have the same absolute value. The symbol |x| represents the absolute value of the number x.

|-8| = 8 |4| = 4 You try: |15| = ? |-23| = ?

We can evaluate expressions that contain absolute value symbols. Think of the | | bars as grouping symbols. Evaluate |9x -3| + 5 if x = -2 |9(-2) -3| + 5 |-18 -3| + 5 |-21| + 5 21+ 5=26

Equations may also contain absolute value expressions When solving an equation, isolate the absolute value expression first. Rewrite the equation as two separate equations. Consider the equation | x | = 3. The equation has two solutions since x can equal 3 or -3. Solve each equation. Always check your solutions. Example: Solve |x + 8| = 3 x + 8 = 3 and x + 8 = -3 x = -5 x = -11 Check: |x + 8| = 3 |-5 + 8| = 3 |-11 + 8| = 3 |3| = 3 |-3| = 3 3 = 3 3 = 3

Now Try These Solve |y + 4| - 3 = 0 |y + 4| = 3 You must first isolate the variable by adding 3 to both sides. Write the two separate equations. y + 4 = 3 & y + 4 = -3 y = -1 y = -7 Check: |y + 4| - 3 = 0 |-1 + 4| -3 = 0 |-7 + 4| - 3 = 0 |-3| - 3 = 0 |-3| - 3 = 0 3 - 3 = 0 3 - 3 = 0 0 = 0 0 = 0

Absolute value is never negative. Therefore, there are no solutions! |3d - 9| + 6 = 0 First isolate the variable by subtracting 6 from both sides. |3d - 9| = -6 There is no need to go any further with this problem! Absolute value is never negative. Therefore, there are no solutions!

Solve: 3|x - 5| = 12 |x - 5| = 4 x - 5 = 4 and x - 5 = -4 x = 9 x = 1 Check: 3|x - 5| = 12 3|9 - 5| = 12 3|1 - 5| = 12 3|4| = 12 3|-4| = 12 3(4) = 12 3(4) = 12 12 = 12 12 = 12

Solve: |8 + 5a| = 14 - a 8 + 5a = 14 - a and 8 + 5a = -(14 – a) Set up your 2 equations, but make sure to negate the entire right side of the second equation. 8 + 5a = 14 - a and 8 + 5a = -14 + a 6a = 6 4a = -22 a = 1 a = -5.5 Check: |8 + 5a| = 14 - a |8 + 5(1)| = 14 - 1 |8 + 5(-5.5) = 14 - (-5.5) |13| = 13 |-19.5| = 19.5 13 = 13 19.5 = 19.5