Question 594536: Solving Inequalities with Variables on both sides
The question is
23-12x>-(7+2x)
Please help me I keep getting the wrong answers each time. I need this by tonight.
Found 2 solutions by jim_thompson5910, jsmallt9:
Answer by jim_thompson5910(35256)   (Show Source):
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! 23 - 12x > -(7+2x)
When solving equations or inequalities, start by simplifying each side. The left side is already simplified. But on the right side we can distribute the "-" (which is short for -1):
23 - 12x > -7 + (-2x)
For many reasons I like to change subtractions to equivalent additions:
23 + (-12x) > -7 + (-2x)
The next step is the make sure the variable is on just one side of the equation/inequality. Here's a helpful hint: Keep the variable on the side of the equation/inequality where it has the larger coefficient (number in front). (I'll explain later why this is a good idea.).
So which coefficient is larger, -12 or -2? Careful. It's easy to get this wrong. -2 is larger than -12! So we want to keep the variable on the right side (where it has its larger coefficient). IOW, we want to get rid of the -12x. We can do this by adding 12x to each side:
23 > -7 + 10x
Next we want to eliminate any extra terms on the side where the variable is. This means the -7 needs to go. Adding 7 to each side we get:
30 > 10x
Next we divide by 10:
3 > x
The only thing left is to read this correctly. For equations the left and right sides are equal and can be swapped back and forth if desired. But this cannot be done with inequalities.
One should learn how to read inequalities. Read inequalities starting from where the variable is. Since the variable is on the right side of 3 > x, the we should read this from right to left. (This skill will take practice since we are used to reading from left to right in English (and many other languages).)
3 > x is read "x is less than 3". So all numbers less than 3 are solutions to your inequality.
Let's take a look at what can happen if one does not use the hint mentioned above.
23 + (-12x) > -7 + (-2x)
Instead of keeping the x on the right side where the larger coefficient is, let's keep the x on the left. Adding 2x to each side we get:
23 + (-10x) > -7
Subtracting 23 from each side we get:
-10x > -30
Now we divide by -10. (This is the problem. First, negatives are a little harder to deal with than positives. Second, and more important, is that there is a special rule for dividing (or multiplying) both sides of and inequality by a negative number! This rule is easy to forget. If one uses my hint, you never have to divide by a negative so you never have to worry about this rule.
The rule for dividing (or multiplying) both sides of an inequality by a negative number is that you must also reverse the inequality. So when we divide both sides by -10 we get:
x < 3
which (reading this one from left to right because the variable is on the left) says "x is less than 3" just like the earlier solution.
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