Why have inequalities at all? Here are some examples of real life inequality situations -- mathematical ones, anyway. Most of us don't set up word problems when we think through them. We just make do. I only worry how to solve what is presented to me.
Second grade (guessing) math sign review
x < = less than but not including
x > = greater than but not including
x < = less than or equal to
x > = greater than or equal to
The "but not including" part isn't usually included when one first learns about greater than and less than. It is crucial to note the difference when graphing inequalities.
Rule #1 for inequalities is reverse the greater/less than sign if you multiply or divide the entire inequality by a negative number. Or the "faster, more intense" version:
REVERSE THE GREATER/LESS THAN SIGN IF YOU MULTIPLY OR DIVIDE THE ENTIRE INEQUALITY BY A NEGATIVE NUMBER.
Basically inequalities are solved the same way as equalities but without the equals sign.
Solve -5x + 3 > -32 for x.
-5x + 3 > -32
-5x + 3 - 3 > -32 + -3
-5x > -35
-x > -7
-1(-x > -7) -- Apply Rule #1
x < 7
Meaning x is any number less than but not including 7. 6.99 will work. 7 won't.
Compound Inequalities
Using only one sign isn't enough? You want to x to be greater than AND less than in the same inequality? Compound equations are for you. Consider it minimum and maximum speeds on the interstate. 45 < x < 70 where x is ticket-free speed. In algebra books x has a coefficient which must be dealt with.
Solve inequality -4 < 3x + 2 < 20
1. Isolate x.
-4 + (-2) < 3x + 2 + (-2) < 20 + (-2)
-6 < 3x < 18
2. Eliminate the coefficient.
(-6)/3 < (3x)/3 < 18/3
-2 < x < 6
On this hypothetical highway you can drive as low as and including -2mph and up to but not including 6mph.
Solving Absolute Value Inequalities
Absolute value is the Jessica Fletcher of algebra. Everything is nice, neat, and tidy in your police report head. Then here comes this well-intentioned mystery writer asking too many questions two little bars which upend all understanding of the murder inequalities. We could also go Columbo here with "just one more thing," but I'm a Murder, She Wrote fan.
But what if the murderer used an absolute value inequality? We use them all the time in Cabot Cove.
(Side note: This still is from "One White Rose for Death," Season 3 Episode 4. For those of us born too late and too far from Broadway to see the original production of Sweeney Todd, this episode is a fantastic opportunity to watch Len Cariou and Angela Lansbury work together. And in a later episode his character plays a priest. No meat pies, though.)
Why are you staring at me like that?
Fine, fine, back to the nerdiness you signed up for.
If the absolute value is on the LESS THAN side:
Solve the inequality |2x - 1| + 3< 6
1. Isolate absolute value portion on the left side of inequality. If it's not already there, rewrite it that way (e.g. -- 6 > |2x - 1| + 3 = |2x - 1| + 3 < 6). At least that's what Idiot's Guide says. It may not matter that much.
|2x - 1| + 3 - 3 < 6 - 3
|2x - 1| < 3
2. Drop the absolute value bars and create a compound inequality with the opposite of the "greater than" integer. If you set things up correctly it's the number to the right of the < sign.
-3 < 2x - 1 < 3
3. Solve the compound inequality
-3 + 1 < 2x - 1 + 1 < 3 + 1
-2 < 2x < 4
-1 < x < 2
Even though this example uses <, < uses the same process.
If the absolute value is on the GREATER THAN side:
Here's where it turns out there are two murderers answers. It gets complicated. Stay with me until the trap is sprung.
Solve the inequality |2x + 5| - 4 > -1
1. Isolate the absolute value just as you did with "less than."
|2x + 5| - 4 > -1
|2x + 5| - 4 + 4 > -1 + 4
|2x + 5| > 3
2. Split this into TWO inequality statements. One simply drops the absolute value bars. The other one drops the absolute value bars, flips the sign to "less than," and negates the integer. Put an "or" between them.
2x + 5 > 3 OR 2x + 5 < -3
3. Solve each inequality separately.
2x + 5 - 5 > 3 - 5 OR 2x + 5 - 5 < -3 - 5
2x > -2 OR 2x < -8
x > -1 OR x < -4
In summation:
|x + a| < b = ONE answer in the form of a compound inequality
|x + a| > b = TWO answers with an OR statement between them
Graphing inequalities
If graphing on a number line open dot means "but not including," filled dot means "and is equal to."
If graphing on a coordinate plane, dashed line means "but not including," solid line means "and is equal to."
Absolute value inequalities will have V-shaped graphs, not lines.
I have no desire to draw examples in Photoshop. Go look at someone else's work. Number line, coordinate plane
Tomorrow I will explain matrices as best I can.
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