Download Category 5 - Algebra - Meet #3 – Study Guide

Category 5 - Algebra - Meet #3 – Study Guide
Absolute Value; inequalities in one variable including interpreting line graphs
Inequalities : Solving inequalities is exactly the same as solving equations with ONE
EXCEPTION!!!!!! If at anytime in solving an inequality you need to multiply or divide
both sides of the inequality by a negative number, you must reverse the < or >.
ex. Solve
x+5
≤ −4
−2
×–2 ×–2
5 – 2x > 13
–5
–5
– 2x > 8
÷–2 ÷–2
x+5≥8
- 2 -2
x≥6
x<–4
Graphing Inequalities :
x < 3 has a graph like
The empty circle shows x ≠ 3
0
3
x ≥ – 4 has a graph like
Filled in circle means x could = -4
–4
0
Absolute Value takes a value and tells you its distance from zero, or takes a DIFFERENCE
and tells you the distance between the values you were taking the difference of.
ex. − 4 since – 4 is 4 away from zero, the absolute value of – 4 is 4
ex. 5 since 5 is 5 away from zero, the absolute value of 5 is 5
ex. 0 since zero is zero away from zero the absolute value of zero is 0
ex. x − 3 = 6 is asking what numbers(x) are 6 away from 3. So 3 + 6 = 9 & 3 – 6 = – 3
You can be more formal about this by writing 2 separate equations :
x – 3 = 6 or x – 3 = – 6 when you add 3 to both sides of each equation you still get x = 9
or – 3
ex. 2 x − 5 = 11
write two equations :
2x – 5 = 11
+5 +5
2x = 16 or
÷2 ÷2
x=8
or
or 2x – 5 = – 11
+5
+5
2x = – 6
÷2
÷2
x= –3
When faced with an absolute value inequality it is most important that you do two things.
The first is to think about what the inequality MEANS. The second is to check your
solution to see if it makes sense.
Using the absolute value equation above as an example and making it an inequality keeps the
solution similar in come ways and different in others.
First, what does 2 x − 5 < 11 mean? It means the value of 2x – 5 must have an absolute value
less than 11. Well, what numbers have an absolute value less than 11? Numbers less than
11 do, but only up until they become less than -11. For example. – 12 has an absolute
value of 12 which is not less than 11.
So, what can 2x – 5 be? 2x – 5 < 11 AND 2x – 5 > – 11 (the AND means x must fulfill
both inequalities)
+5 +5
+5 +5
2x < 16 AND
2x > – 6
÷2 ÷2
÷2
÷2
x<8
AND
x> –3
That tells us x can be anything between 8 and – 3 .
If asked for the integer values of x, that would be {7, 6, 5, 4, 3, 2, 1, 0, -1, -2}
Now, CHECK your answers!!!! The 8 and – 3 should be the value of x that make the original
an equation, not an inequality 2(8) − 5 = 11 and 2(-3) - 5 = 11 so that's good. What about
numbers between them?
2(4) − 5 = 3 and 2(-1) - 5 = 7 good, they both came out less than 11 as they should. What about
numbers outside that range? 2(20) − 5 = 35 and 2(-10) - 5 = 25 Great those were not
supposed to work, and they didn't, they gave values outside the 8 to – 3 range!!
Some problems will actually be easier to do starting by "guessing" right away, but only if you
understand what you're doing and how absolute values work!!