1. Get out your notebooks! Today’s Objectives:
You will be able to solve absolute value equations.
Today’s Objectives:

2. Solve the following equations. −2 3𝑥−5 −4 𝑥+10 =−50 3 𝑥−9 −2 6𝑥−3 =6
Warm Up
Solve the following equations.
−2 3𝑥−5 −4 𝑥+10 =−50
3 𝑥−9 −2 6𝑥−3 =6

3. 1.4 Solving Absolute Values
Find the distance between the following two points on the real number line. -5 -4 -3 -2 -1 1 2 3 4 5
2 𝑎𝑛𝑑 −5
−3 𝑎𝑛𝑑 −1
−5 𝑎𝑛𝑑 2
1 𝑎𝑛𝑑−3

4. Distance
The distance between two real numbers is the absolute value of their ___________.
2 𝑎𝑛𝑑 −5
−3 𝑎𝑛𝑑 −1
−5 𝑎𝑛𝑑 2
1 𝑎𝑛𝑑−3

5. Distance Statements
Rewrite the absolute value statements as distance statements.
3−2 is the same as saying “the distance between ____ and ____.”
5+2 is the same as saying “the distance between ____ and ____.”

6. Distance Statements
Rewrite the absolute value statements as distance statements.
𝑥−4 is the same as saying “the distance between ____ and ____.”
𝑥 is the same as saying “the distance between ____ and ____.”

7. Distance
Therefore, distances are always positive because an absolute value is always positive.

8. Scenario
Sarah’s height is "𝒔" inches and Jessica’s height is "𝒋" inches.
Their father wants to know how many inches separate his two daughters.
Write an equation for this difference in such a way that the result will always be positive no matter which sister is taller.

9. Copy This Down…
Sarah is at mile marker 44. She is 7 miles from her exit. At what mile marker could Sarah’s exit be located?

10. Scenario
Make a visual representation on a number line of “the distance from mile marker 44 to Sarah’s exit x is 7.”
Expanded solution:
How might we write a mathematical equation for this visual representation?

11. words: the distance from 𝑥 to 0 is _____.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥−0 =2
words: the distance from 𝑥 to 0 is _____.
𝒙=______ or 𝒙=______ -5 -4 -3 -2 -1 1 2 3 4 5

12. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥 =5
words: _____________________________.
𝒙=______ or 𝒙=______ -5 -4 -3 -2 -1 1 2 3 4 5

13. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥−1 =3
words: _____________________________.
𝒙=______ or 𝒙=______ -5 -4 -3 -2 -1 1 2 3 4 5

14. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
1−𝑥 =3
words: _____________________________.
𝒙=______ or 𝒙=______ -5 -4 -3 -2 -1 1 2 3 4 5

15. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥+1 =4
words: _____________________________.
𝒙=______ or 𝒙=______ -5 -4 -3 -2 -1 1 2 3 4 5

16. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥+2 =3
words: _____________________________.
𝒙=______ or 𝒙=______

17. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥−5 =0
words: _____________________________.
𝒙=______ or 𝒙=______

18. words: _____________________________.
Examples
Write a sentence in words, draw a graph, and identify the solutions.
𝑥+3 =−4
words: _____________________________.
𝒙=______ or 𝒙=______

19. Generalizing 𝑥−𝑏 =𝑐 words: _______________________________.
Which value (𝑏 or 𝑐) is plotted first?_____
Which value tells the difference?_____
𝒙=______ or 𝒙=______

20. Word Problem
Identical vacation cottages, equally spaced along a street, are numbered consecutively. Maria lives in cottage #17. Joshua lives 4 cottages away from Maria. If 𝑛 represents Joshua’s cottage number, then write an inequality that expresses this situation. Draw a visual representation on a number line, and solve the inequality.

21. 2 =22 2 𝑥−6 =22 𝒙=______ or 𝒙=______
Examples
2 =22
2 𝑥−6 =22
𝒙=______ or 𝒙=______

22. 9+3 =12 9+3 𝑥+8 =12 𝒙=______ or 𝒙=______
Examples
=12
9+3 𝑥+8 =12
𝒙=______ or 𝒙=______

23. 4𝑥−2 =10 4𝑥 𝟒𝒙=______ or 𝟒𝒙=______ 𝒙=______ or 𝒙=______
Examples
4𝑥−2 =10 4𝑥
𝟒𝒙=______ or 𝟒𝒙=______
𝒙=______ or 𝒙=______

24. 200−3𝑥 =896 3𝑥 𝟑𝒙=______ or 𝟑𝒙=______ 𝒙=______ or 𝒙=______
Examples
200−3𝑥 =896 3𝑥
𝟑𝒙=______ or 𝟑𝒙=______
𝒙=______ or 𝒙=______

25. Examples
2𝑥+3 =5

26. Examples
70+5𝑥 =15

27. Examples
3 2𝑥−3 −5=4

28. Examples
−2 5𝑦−1 =−48

29. Ticket Out The Door
Write an absolute value equation that corresponds to…
Draw a visual representation on a number line, and
Solve the equation.
The carnival guesser tried to guess Aiden’s weight, but she was off by 9 pounds. Aiden actually weights 168 lbs. There are two guesses she could have made, what are they?

30. Homework
Lesson 1.4
Pg. 30 #’s 23-24, 27-32