Solving Absolute Value Equations Inequalities Absolute Value of

Solving Absolute Value Equations & Inequalities

Absolute Value (of x) • Symbol lxl • The distance x is from 0 on the number line. • Always positive • Ex: l-3 l=3 -4 -3 -2 -1 0 1 2

Ex: x = 5 • What are the possible values of x? x=5 or x = -5

To solve an absolute value equation: ax+b = c, where c>0 To solve, set up 2 new equations, then solve each equation. ax+b = c or ax+b = -c ** make sure the absolute value is by itself before you split to solve.

Ex: Solve 6 x-3 = 15 6 x - 3 = 15 or +3 +3 6 x = 18 or 6 6 x=3 or 6 x - 3 = -15 +3 +3 6 x = -12 6 6 x = -2 * Plug in answers to check your solutions!

Ex: Solve 2 x + 7 -3 = 8 Get the abs value alone by adding 3! 2 x + 7 = 11 Now split into 2 parts. 2 x+7 = 11 or 2 x+7 = -11 2 x = 4 or 2 x = -18 x = 2 or x = -9 Check the solutions.

6|5 x + 2| = 312 • Isolate the absolute value expression by dividing by 6. 6|5 x + 2| = 312 |5 x + 2| = 52 • Set up two equations to solve. 5 x + 2 = 52 5 x = 50 x = 10 or 5 x + 2 = -52 5 x = -54 x = -10. 8 • Check: 6|5 x + 2| = 312 6|5(10)+2| = 312 6|52| = 312 6|5 x + 2| = 312 6|5(-10. 8)+ 2| = 312 6|-52| = 312

3|x + 2| -7 = 14 Isolate the absolute value expression by adding 7 and dividing by 3. 3|x + 2| -7 = 14 +7 +7 3|x + 2| = 21 3 3 |x + 2| = 7 Set up two equations to solve. x+2=7 x + 2 = -7 x=5 or x = -9 • Check: 3|x + 2| - 7 = 14 3|x + 2| -7 = 14 3|5 + 2| - 7 = 14 3|-9+ 2| -7 = 14 3|7| - 7 = 14 3|-7| -7 = 14 21 - 7 = 14 14 = 14

Solving Absolute Value Inequalities 1. ax+b < c, where c>0 Becomes an “and” problem Changes to: –c<ax+b<c 2. ax+b > c, where c>0 Becomes an “or” problem Changes to: ax+b>c or ax+b<-c In other words, change All Signs!

Ex: Solve & graph. • Becomes an “and” problem -3 7 8

Solve & graph. • Get absolute value by itself first. • Becomes an “or” problem -2 3 4

Lesson : Adding Integers Using Rules Taught by the one and only Mr. Richard, Teacher most Extraordinaire, and delivered to his very Awesome

Adding Integers We are going to predict patterns and make rules based on what we notice. Then, we are going to simplify variable expressions.

Adding Integers Solve the problems by using + and – to show your work. 1) 4 + 5 2) 16 + 13 Predict: Predict What do you notice about your answers? What rule can we make?

We can show this same idea using a number line. 9 -10 -5 0 5 What is 5 + 4? Move five (5) units to the right from zero. Now move four more units to the right. The final point is at 9 on the number line. Therefore, 5 + 4 = 9. 10

Adding Integers Rule 1: 1 Adding positives will give you a positive. + + +=+

Adding Integers Solve the problems by using + and – to show your work. 1) -12 + (-3) 2) -5 + (-17) Predict: Predict What do you notice about your answers? What rule can we make?

-9 -10 -5 0 5 What is -5 + (-4)? Move five (5) units to the left from zero. Now move four more units to the left. The final point is at -9 on the number line. Therefore, -5 + (-4) = -9. 10

Adding Integers Rule 2: Adding negatives will give you a negative. - +-=-

What is (-7) + 7? -5 0 5 -10 10 To show this, we can begin at zero and move seven units to the left. Now, move seven units to the right. Notice, we are back at zero (0). For every positive integer on the number line, there is a corresponding negative integer. These integer pairs are opposites or additive inverses. Additive Inverse Property – For every number a, a + (-a) = 0

Add the following integers: -10 -5 (-4) + 7. 0 Start at zero and move four units to the left. Now move seven units to the right. The final position is at 3. Therefore, (-4) + 7 = 3. 5 10

Add the following integers: -10 -5 (-4) + 7. 0 5 Notice that seven minus four also equals three. In our example, the number with the larger absolute value was positive and our solution was positive. Let’s try another one. 10

Add (-9) + 3 -10 -5 0 Start at zero and move nine places to the left. Now move three places to the right. The final position is at negative six, (-6). Therefore, (-9) + 3 = -6. 5 10

Add (-9) + 3 -10 -5 0 5 10 In this example, the number with the larger absolute value is negative. The number with the smaller absolute value is positive. We know that 9 -3 = 6. However, (-9) + 3 = -6. 6 and – 6 are opposites. Comparing these two examples shows us that the answer will have the same sign as the number with the larger absolute value.

To add integers with different signs determine the absolute value of the two numbers. Subtract the smaller absolute value from the larger absolute value. The solution will have the same sign as the number with the larger absolute value. Example Subtract Solution

Adding Integers Rule 3: Adding a BIGGER positive and smaller negative will give you a positive. TAKE THE SIGN OF THE BIGGER NUMBER. ++ - = +

Adding Integers Rule 4: 4 Adding a smaller positive and BIGGER negative will give you a NEGATIVE. TAKE THE SIGN OF THE BIGGER NUMBER. + + -=-

Adding Integers Solve. 1) -9 + 14 2) -7 + -2 3) 4 + (-11) 4) 6 + (-2) + 3 + (-8)

Additional Example 3: Evaluating Expressions with Integers Evaluate c + 4 for c = – 8. c+4 (– 8) + 4 Replace c with – 8. Think: Find the difference of – 8 and 4. – 8 + 4 = – 4 8 > 4; use the sign of 8.

Check It Out! Example 3 Evaluate d + 7 for d = – 3. d+7 (– 3) + 7 Replace d with – 3. Think: Find the difference of – 3 and 7. – 3 + 7 = 4 7 > 3; use the sign of 7.

Additional Example 4: Application Meka opened a new bank account. Find her account balance after the first four transactions listed below. Deposits: $200, $20 Withdrawals: $166, $38 200 + 20 + (– 166) + (– 38) Use a positive sign for deposits and a negative sign for withdrawals. Group integers with the (200 + 20) + (– 166 + – 38) same signs. 220 + (– 204) Add integers within each group. 220 > 204; use the sign 16 of 220. Meka’s account balance after the first four transactions is $16.

Check It Out! Example 4 Michael opened a new bank account. Find his account balance after the first four transactions listed below. Deposits: $175, $95 Withdrawals: $133, $65 175 + 95 + (– 133) + (– 65) (175 + 95) + (– 133 + – 65) 270 + (– 198) 72 Use a positive sign for deposits and a negative sign for withdrawals. Group integers with the same signs. Add integers within each group. 270 > 198; use the sign of 270. Michael’s account balance after the first four transactions is $72.

Add. Lesson Quiz 1. – 7 + (– 7) – 14 2. 15 + (– 9) + (– 2) 4 Evaluate each expression for the given value of the variable. – 2 3. 13 + r for r = – 15 – 13 4. 2 + b + (– 9) for b = – 6 5. On Monday, a local dog shelter had six dogs. By Friday, they had found homes for three, but took in two more. Then how many dogs were in 5 dogs the shelter?