SOLUTION: Hi, I have no idea how to do this classwork problem. COuld you help? Thanks! Solve for x. Collect xs on the LEFT side. The solution should be presented with x first, th

Question 77063: Hi,
I have no idea how to do this classwork problem. COuld you help? Thanks!

Solve for x. Collect xs on the LEFT side. The solution should be presented with x first, the correct inequality symbol, then the number.
x + 4 � 8x � 1 > 3(x � 2)

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
You can work on inequalities using nearly the same methods as you do on equations. One
important difference is that if you multiply or divide both sides of an inequality by a
negative quantity, then you must reverse the direction of the inequality arrow. With these
ideas in mind, let's work your problem so that you can see these methods in use.
.
Given: x + 4 � 8x � 1 > 3(x � 2)
.
Let's multiply out the right side as a first step. When you do the distributed multiplication
on the right side the inequality becomes:
.
x + 4 � 8x � 1 > 3x - 6
.
Now just as we would in an equation, we're going to collect the terms that contain x
on the left side and the numbers on the right side of the inequality sign.
.
Let's start by combining the x and the -8x on the left side. They combine to give you
-7x and the inequality then becomes:
.
-7x + 4 - 1 > 3x - 6
.
Next, on the left side the +4 and -1 can be added to give +3. So the inequality then
becomes:
.
-7x + 3 > 3x - 6
.
Because we want only terms containing x on the left side, we need to get rid of the +3
on the left side. Do that by subtracting 3 from the left side. But, just like in an
equation, if you subtract 3 from the left side, you must also subtract 3 from the right
side. After you subtract 3 from both sides, the inequality is:
.
-7x > 3x - 9
.
[The -9 comes from - 6 - 3 = -9.]
.
Now, we need to get rid of the 3x on the right side by subtracting 3x from the right
side. And we then have to also subtract 3x from the left side. After subtracting
3x from both sides the inequality then is:
.
-10x > -9
.
Almost done. Remember that we are solving for +x. We can do that by dividing both sides
of this inequality by -10 ... the multiplier of the x on the left side. When you do that
division you get:
.
x > 9/10
.
But we divided both sides by a negative number. And the rule from above is that when you
divide both sides by a negative number, you must reverse the direction of the inequality
arrow. If you reverse the arrow, the answer becomes:
.
x < 9/10
.
or in decimal form:
.
x < 0.9
.
It pays to check try a couple of checks just to build our confidence that this answer
is correct. Let's try a value of x that is less than 0.9 ... say x = 0. Go back to the
original problem and set x equal to zero and see if the result is correct. The original
problem said:
.
x + 4 � 8x � 1 > 3(x � 2)
.
and when x is zero this reduces to:
.
+4 - 1 > 3(-2)
.
The left side combines to +3 and the right side multiplies to -6 so that the inequality
becomes:
.
+ 3 > -6
.
Since +3 is to the right of -6 on the number line, clearly +3 is greater than -6 and the
inequality is correct.
.
Now let's try a value of x that is greater than 0.9 ... let's try x = 1. Substitute
+1 for x in the original problem:
.
x + 4 � 8x � 1 > 3(x � 2)
.
1 + 4 - 8 - 1 > 3(1 - 2)
.
The left side algebraically adds to -4 and the right side multiplies to -3. This means
when x = 1 the inequality becomes:
.
-4 > -3
.
That is not true. -4 is not to the right of -3 on the number line. Therefore, minus 4 is
actually less than -3.
.
These two spot checks make us more confident that x must be less than 0.9 ... when x
was 0 (which is less than 0.9) it made the original inequality work correctly, and when
x was 1 (which is greater than 0.9) it made the original inequality incorrect.
.
Hope this helps you to understand inequalities a little better. You can plan on solving
for the positive unknown (in this case +x) by following the same general procedures
that you use in solving an equation ... with the exception of that rule that says if you
multiply or divide both sides by a negative number you need to reverse the direction
of the inequality arrow.

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