4.5 solving absolute value inequalities 10/30/13
TRANSCRIPT
4.5 Solving Absolute ValueInequalities
10/30/13
or
Solve an “Or” Compound Inequality
Solve or
From 4.1: Compound Inequlalities
Solve an “And” Compound Inequality
Solve
SOLUTION
Solve |x| < 3
What value of x would make what’s inside | | 7 or
-7 What numbers are less than 3 units away from 0?
Answer: Numbers to the right of -3, like -2, -1 and anything to the left of 3 like 2, 1
Mathematically written: -3 < x < 3
x > -3
x < 3
Solve |x| ≤ 3
Answer: Numbers to the right of -3 including 3 and anything to the left of 3 including 3.
Mathematically written: -3 ≤ x ≤ 3
x ≥ -3
x ≤ 3
Solve |x| > 4
What value of x would make what’s inside | | 7 or
-7 What numbers are more than 4 units away from 0?
Answer: Numbers to the right of 4, like 5, 6, 7 or anything to the left of - 4 like -5, -6, -7
Mathematically written: x < -4 or x > 4
x > 4
x < -4
Solve |x| ≥ 4
Answer: Anything to the right of 4 including 4 or anything to the left of - 4 including -4
Mathematically written: x ≤ -4 or x ≥ 4
x ≥ 4
x ≤ -4
< and ≤ are both AND problems> and ≥ are both OR problems (think greater pronounced as greatOR)
“AND”
“OR”
Original Problem Solutions: Graph of solution|x| < 3 -3 < x < 3 Shade between
the 2 numbers|x| ≤ 3 -3 ≤ x ≤ 3
|x| > 4 x > 4 or x < -4
|x| ≥ 4 x ≥ 4 or x ≤ -4
Shade opposite direction (away from each other) from 4 and -4Shade opposite
direction (away from each other)
|𝑎𝑥+𝑏|<𝑐
|𝑎𝑥+𝑏|≤𝑐
−𝑐<𝑎𝑥+𝑏<𝑐
−𝑐 ≤𝑎𝑥+𝑏≤𝑐
Solving Absolute Value Inequalities
Original Problem Rewrite as Graph of solution
|𝑎𝑥+𝑏|>𝑐 or
|𝑎𝑥+𝑏|≥𝑐 or
Example 1 Solve an Inequality of the Form
Solve . Then graph the solution.
+x b ≤ c
+x 4 ≤ 10
– +x 4 ≤ 10 Write as “AND” compound inequality.10 ≤
– x ≤ 614 ≤
16–.14– 12– 10– 8– 6– 4– 2– 0 2 4 6 8
.
+16 4– 10≤ +0 4 10≤ +8 4 10≤
12 10≤ 4 10≤ 12 10≤
CHECK Test one value from each region of the graph.
SOLUTION
-4 - 4 -4
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Absolute Value Inequality
1. +x 1 ≤ 4
ANSWER – x ≤ 35 ≤
6– 4– 2– 0 2 4. .3
Example 2 Solve an Inequality of the Form
Solve . Then graph the solution.
+ax b c<
+2x 3 7<
– Write as “AND” compound inequality.+2x 3 7<7 <
–10 2x 4<<
–5 x 2<<
6– 5– 4– 3– 0 1 22– 1– 3
SOLUTION
-3 - 3 -3
2 2 2
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Absolute Value Inequality
2. 1 4x 3<+ANSWER
1– 0..
– x 0.51 < <
13.
Example 3 Solve an Inequality of the Form
Solve . Then graph the solution.
+ax b c≥
x 1 32
1– ≥
FIRST INEQUALITY SECOND INEQUALITY
x 1 32
1– ≤ – x 1 3
2
1≥–
Rewrite as “OR” Compound Inequalities
+1 +1
+1 +1
8
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Absolute Value Inequality
4. 2x 3 7+ >
5. 4x 1 5+ ≥
ANSWER –1.5 orx x 1≥
1– 0 12–. .
≤
ANSWER < –5 orx 2>x
4– 2– 0 2 4 66–
5–
Homework: WS 4.5
Quiz 4.3-4.5 Friday
“I tried to catch some fog. I mist.”