SOLUTION: How do you work these problems? I have no Idea where to start! Help! Solve and graph the solution: -12 is greater than or equal to 24x

Algebra ->  Inequalities -> SOLUTION: How do you work these problems? I have no Idea where to start! Help! Solve and graph the solution: -12 is greater than or equal to 24x      Log On


   

Question 93529: How do you work these problems? I have no Idea where to start! Help!
Solve and graph the solution:
-12 is greater than or equal to 24x
Found 2 solutions by bucky, jim_thompson5910:
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
In equation form you are given:
.
-12%3E=24x
.
You can pretty much work these using the same procedures that you would to solve an equation.
.
However, there is an exception. This exception is that in solving for x if you ever have to
MULTIPLY or DIVIDE both sides by a NEGATIVE quantity, then you also have to reverse the
direction of the inequality sign.
.
Let's work this problem.
.
If this were an equation, you would solve for x by dividing both sides by 24 because 24
is the multiplier of x. So let's divide both sides by +24 to get:
.
-12%2F24%3E=%2824x%29%2F24
.
Notice that on the left side if you divide both the numerator and denominator by 12 you are
left with -1%2F2 and on the right side the 24 in the numerator cancels with the 24
in the denominator. Therefore, all you are left with is:
.
-1%2F2%3E=x
.
That's the answer. Now, what does it mean? It says that -1%2F2 is greater than or equal
to x. (Remember that the inequality sign always points to the smaller quantity.)
.
This also means the same as x is less than or equal to -1%2F2. For the purposes of
graphing this answer, you can draw a number line. Then put a dot at the point -1%2F2.
x can have that value. x can also be less than that number. That means that x can be any
value to the left of the point at -1%2F2. So you make the number line to the left of the
point -1%2F2 a very heavy line to indicate that x can be on that part of the line.
.
Hopefully this makes some sense to you. In solving this problem we didn't have to multiply
or divide both sides by a negative number, so we didn't need to use the rule about
changing the direction of the inequality sign. It just stayed the same direction
throughout all our work.
.
Remember the goal is to solve inequalities for +x. If you have a problem that you can solve
for -x, then do so. But at the very end of the problem you will need to multiply the -x by
-1 to change it to +x. And when you multiply one side by -1 you must also multiply the
other side by -1 ... and that means that you also need to reverse the direction of the
inequality sign because you multiplied both sides by a minus number.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
-12%3E=24x

24x%3C=-12 Flip the sides so the variable is on the left side

x%3C=%28-12%29%2F%2824%29 Divide both sides by 24 to isolate x

x%3C=-1+%2F+2 Reduce

So our answer is x%3C=-1+%2F+2 or x%3C=-0.5 in decimal form


Start with the given inequality:

x%3C=-0.5

Now lets graph x%3C=-0.5



Start with the given inequality:

x%3C=-0.5

Now let's graph x%3C=-0.5


Set up a number line:
number_line%28500%2C-10.5%2C9.5%29

Now plot the point x=-0.5 on the number line


number_line%28500%2C-10.5%2C9.5%2C+-0.5%29


Now pick any test point you want, I'm going to choose x=0, and test the inequality x%3C=-0.5


0%3C=-0.5 Plug in x=0


Since this inequality is not true, we simply shade the entire portion that does not contain the point x=0 using the point x=-0.5 as the boundary. This means we shade everything to the left of the point x=-0.5 like this:
Graph of x%3C=-0.5 with the shaded region in blue
note: at the point x=-0.5, there is an closed circle. This means the point x=-0.5 is included from the solution set.