Question 73747
Given:
.
{{{abs(c-4)>1}}}
.
solve and graph
.
One way to do absolute value problems is to solve them as two separate problems.  For the
first problem, remove the absolute value signs and solve:
.
+(c-4)>1
.
You can treat this just as you would and equation ... except for one difference. That difference
is that if you have to multiply or divide both sides by a negative number, you must then
reverse the direction of the inequality sign. Don't forget that.
.
Now back to solving:
.
+(c-4)>1
.
The parentheses are preceded by a plus sign so you can just remove them and the inequality
becomes:
.
+c - 4 > 1
.
Just as you would in an equation, eliminate the -4 on the left side by adding 4 to both
sides to get:
.
+c > 5
.
That's the first limit on the value of c ... it must be greater than +5.
.
Now, on to the second equation.  We do the same thing as before ... remove the absolute
value signs, except this time we put a minus sign in front of the quantity that was inside
the absolute value signs.  When you do this you have:
.
-(c-4)>1
.
Because the quantity is preceded by a negative sign, when you remove the parentheses
you must change the sign of every term that was inside the parentheses.  When you do
that the inequality becomes:
.
-c + 4 >1
.
We need to solve this for +c.  Let's begin by eliminating the +4 on the left side by
subtracting 4 from both sides.  This causes the inequality to become:
.
-c > -3
.
To change the equation so that the left side is +c, multiply both sides by negative 1.
And when you do, do NOT forget to reverse the direction of the inequality sign to get:
.
+c < +3
.
This tells us that c must be less than +3
.
Combined, the two restrictions are that c can be any number less than 3 and it can also
be any number greater than 5.  But it cannot be either 3 or 5 or any number between those
two values.
.
Let's check that out with a trio of trials. Start with the problem:
.
{{{abs(c-4)>1}}}
.
Let's set c equal to zero.  That's a number less than +3 so it should work. When you substitute
0 for c you get:
.
{{{abs(0-4)>1}}} which simplifies to:
.
{{{abs(-4)>1}}} and since {{{abs(-4) = +4}}} we get +4 > 1.  That works!
.
Now let's set c equal to 4.  That should NOT work.  Substitute 4 for c and get:
.
{{{abs(4-4)>1}}} this obviously becomes:
.
{{{abs(0)>1}}} and the absolute value of 0 is 0.  Obviously 0 > 1 is NOT true so numbers 
between 3 and 4 probably all need to be excluded.  
.
Finally try c = 6.  That is greater than +5 and should work. Substitute +5 for c and get:
.
{{{abs(6-4)>1}}} which becomes {{{abs(6-4)>1}}} and simplifies to {{{abs(2)>1}}}.
At this point it is obvious that 2 > 1 is true and that adds to the likelihood that numbers
greater than 5 will work.
.
We have a good solution!
.
To graph, just make a number line and put dots at +3 and +5. Exclude those dots and shade
the number line from just to the left of +3 all the way to the left and from just to 
the right of +5 all the way to the right.  c can take any value in the shaded region.
.
Hope this shows you a way to work absolute value inequalities.  Remember the two different
equations to solve (one by using the absolute value quantity with a + sign and the other
by using that quantity with a minus sign preceding it), the need to reverse the direction of 
the inequality sign if multiplying or dividing both sides by a negative number, and to 
always solve for the positive value of the variable. The rest is just algebraic manipulation.
.
Hope this works for you.
.