Name_______________________ Period_________________ Date: __________
Activity 3.2 – Absolute Value Inequalities
Activity 1: Absolute Value Inequalities – find the locations:
Find the distance on the number line below for the following problems:
1. If I live at 1 and I want my new friend to live up to 2 houses away, where could my friend’s house be
located? Show this on the number line below:
Represent this by the absolute value inequality
|𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑟𝑖𝑒𝑛𝑑′ 𝑠 ℎ𝑜𝑢𝑠𝑒 − 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑦 ℎ𝑜𝑢𝑠𝑒| :
|𝑥 − 1| ≤ 2
What does the solution set look like?
You can write this as a compound inequality. −2 ≤ 𝑥 − 1 ≤ 2
Solve these:
𝑥 − 1 ≥ −2 𝐚𝐧𝐝 𝑥 − 1 ≤ 2
2. If I live at 1 and I want my new friend to live at most 8 houses away, where could my friend’s house
be located? Show this on the number line below:
Represent this by the absolute value inequality
|𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑟𝑖𝑒𝑛𝑑′ 𝑠 ℎ𝑜𝑢𝑠𝑒 − 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑦 ℎ𝑜𝑢𝑠𝑒| :
|
|≤
What does the solution set look like?
You can write this as a compound inequality. −8 ≤ 𝑥 − 1 ≤ 8
Solve these:
𝑥 − 1 ≥ −8 𝐚𝐧𝐝 𝑥 − 1 ≤ 8
1
3. If I live at −3 and I want my new friend to live at most 4 houses away, where could my friend’s
house be located? Show this on the number line below:
Represent this by the absolute value inequality
|𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑟𝑖𝑒𝑛𝑑′ 𝑠 ℎ𝑜𝑢𝑠𝑒 − 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑦 ℎ𝑜𝑢𝑠𝑒| : BE CAREFUL WITH THE SIGNS:
|
|≤
What does the solution set look like?
Write the compound inequality for this and solve it.
4. I live at −1 and I want my old boyfriend to live no closer than 6 houses away, where could my old
boyfriend’s house be located? Show this on the number line below:
Represent this by the absolute value inequality
|𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑟𝑖𝑒𝑛𝑑′ 𝑠 ℎ𝑜𝑢𝑠𝑒 − 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑦 ℎ𝑜𝑢𝑠𝑒| :
|
|≥
What does the solution set look like?
Another way to write this is: 𝑥 + 1 ≥ 6 𝐨𝐫 𝑥 + 1 ≤ −6
Solve this.
What is the difference between the graph for problem 3 and 4?
Why?
2
5. If I live at −2 and I want my ex-girlfriend to live no closer than 7 houses away, where could my exgirlfriend’s house be located? Show this on the number line below:
Represent this by the absolute value inequality
|𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑟𝑖𝑒𝑛𝑑′ 𝑠 ℎ𝑜𝑢𝑠𝑒 − 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑦 ℎ𝑜𝑢𝑠𝑒| :
|
|≥
What does the solution set look like?
Write the compound inequality for this and solve it.
What is the pattern for the graphs for problems 4 & 5.
6. Sketch the absolute value inequality graphs where I wanted my friend to live close to me.
Which type of inequality shows this type of graph? < 𝒐𝒓 >?
7. Sketch the absolute value inequality graphs where I wanted my ex to NOT live close to me.
Which type of inequality shows this type of graph? < 𝒐𝒓 >?
3
Each absolute-value equation and inequality is rewritten as two equations or two inequalities joined by
and or or. In part c, write the compound inequality.
ax  b  c means
a.
b.
ax  b  c means
a.
b.
c.
ax  b  c means
a.
b.
c.
ax  b  c means
a.
b.
c.
ax  b  c means
a.
b.
c.
Solve the equation. Then graph it on the number line. Pick a point to check with if it is a <. Pick two
points to check with if it is a >.
SHOW ALL WORK and write the compound inequality.
8. |𝑣 + 3| > 1
Compound Inequality _________________________________________________
Check(2):
4
9. |𝑧 − 2| ≤ 5 Compound Inequality _________________________________________________
Check(1):
10. |𝑤 − 6| ≥ 3 Compound Inequality _________________________________________________
Check(2):
11. |2𝑛 + 1| < 9 Compound Inequality _________________________________________________
Check(1):
5
12. |2𝑥 + 3| > 4 Compound Inequality _________________________________________________
Check(2):
13. |5𝑥 + 1| − 8 ≤ 16 - simplify to isolate absolute value prior to writing the
Compound Inequality _______________________________________________
Check(1):
14. |3𝑥 + 2| − 1 ≥ 10 simplify to isolate absolute value prior to writing the
Compound Inequality _______________________________________________
Check(2):
6
15. |3 + 𝑥| + 7 < 10 simplify to isolate absolute value prior to writing the
Compound Inequality _______________________________________________
Check(1):
16. The test scores in your class range from 60 to 100. Write an absolute value inequality describing the
range of test scores and graph it below.
Absolute value Inequality _______________________________________
Compound Inequality _______________________________________________
17. On your basketball team the starting players scoring average are between 8 and 22 points per game.
Write an absolute value inequality describing the scoring averages for the players and graph it
below.
Absolute value Inequality _______________________________________
Compound Inequality _______________________________________________
7
18. |3 − 4𝑥| − 17 < 10 simplify to isolate absolute value prior to writing the
Compound Inequality _______________________________________________
Be careful when solving since you have a −4𝑥.
Check(1):
19. |4𝑥 − 5| < −10 simplify to isolate absolute value prior to writing the
Compound Inequality
_______________________________________________
Check(1):
20. |8𝑥 − 5| + 7 < 5 simplify to isolate absolute value prior to writing the
Compound Inequality
_______________________________________________
Check(1):
8