Download Summer Assignment Review

Summer Assignment Review
Graphing Compound Inequalities
And Absolute Value Inequalities
Compound Inequalities
AND compound inequalities should
create a range of values.
Don’t assume that the given
inequality creates this range.
−2 ≤ 𝑥 ≤ 3
−2 ≥ 𝑥 ≥ 3
𝑥 ≥ −2 𝐴𝑁𝐷 𝑥 ≤ 3
𝑥 ≤ −2 𝐴𝑁𝐷 𝑥 ≥ 3
𝑥 falls between -2 and 3 on the
number line.
𝑥 can not be both to the left of -2
and to the right of 3 on the number
line . There is no solution.
Compound Inequalities
OR compound inequalities should
create two ranges of values.
𝑥 ≤ −2 𝑂𝑅
𝑥≥3
𝑥 is either to the left of -2 or to the
right of 3 on the number line.
AND compound inequalities should
create a range of values.
𝑥 ≥ −2 𝑂𝑅 𝑥 ≤ 3
𝑥 can be anywhere on the number
line. The solution is All Real
Numbers.
Practice
Graph each compound inequality.
1)
−3 < 𝑥 < 4
2)
𝑥 ≥ −3 𝑜𝑟 𝑥 < 3
3)
𝑥 < −4 𝑜𝑟 𝑥 ≥ 2
4)
𝑥 ≤ 0 𝑎𝑛𝑑 𝑥 ≥ 3
5)
𝑥 > −1 𝑎𝑛𝑑 𝑥 ≤ 5
All Real
Numbers
No Solutions
Absolute Value Inequalities
When we say that 𝑥 = 3, we are saying that the distance between x and 0 is 3.
When we say that 𝑥 < 3, we are saying that the distance between x and 0 is less than 3.
The inequalities < and ≤ lead to “and” relationships
−3 < 𝑥 < 3
When we say that 𝑥 > 3, we are saying that the distance between x and 0 is greater than 3.
The inequalities > and ≥ lead to “or” relationships
𝑥 < −3 𝑂𝑅 𝑥 > 3
Solving Absolute Value Inequalities
Case 2 - modified method: Drop the
absolute value bars, flip the inequality
symbol and change the sign of the term
on the right.
2𝑥 − 1 < −7
2𝑥 < −6
𝑥 < −3
Solve and Graph an Abs. Val. Ineq.
Example : Solve 𝑥 − 5 ≥ 7 . Graph your solution.
Step 1: Is it and or or?
This is ≥ so it is “or”. Set up two inequalities.
𝑥−5 ≥7
For the second, flip the inequality and
change the sign of the 7
𝑥−5≥7
𝑥 − 5 ≤ −7
Step 2: Solve both
𝑥 ≤ −2
𝑥 ≥ 12
Step 3: Graph
𝑥 ≥ 12
-3 -2 -1
or
𝑥 ≤ −2
· · · 11 12 13
Solve and Graph an Abs. Val. Ineq.
Example : Solve −4𝑥 − 5 + 3 < 9 . Graph your solution.
Step 1: Is it and or or?
We don’t know yet, get the abs. val. alone.
−4𝑥 − 5 + 3 < 9
−4𝑥 − 5 < 6
Step 2: Solve
Now :
It is < , so it is an “and” inequality. Drop
the abs. val. Bars and put -6 < on the left.
−6 < −4𝑥 − 5 < 6
+5
+5
+5
−1 < −4𝑥 < 11
-4
-4
-4
0.25 > 𝑥 > −2.75
−2.75 < 𝑥 < 0.25
Step 3: Graph
Practice
𝑥 > 5 𝑜𝑟 𝑥 < −11
-12 -11 -10
··· 4
5 6
−5 < 𝑤 < 6
-6 -5 -4 · · · 5 6 7
−0.2 ≤ 𝑚 ≤ 2.6