SOLUTION: Solve each inequality. Write the answer using interval notation.
(a) 2 < |x − 4| < 8
(b) (x2 − 16)(x2 + 4x − 5) > 0
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Question 231026: Solve each inequality. Write the answer using interval notation.
(a) 2 < |x − 4| < 8
(b) (x2 − 16)(x2 + 4x − 5) > 0
Found 2 solutions by edjones, jsmallt9:
Answer by edjones(8007) (Show Source): You can put this solution on YOUR website!
(a)
2 < |x − 4| < 8
.
|x-4|>2
x-4>2
x>6
.
x-4<-2
x<2
.
|x-4|<8
x-4<8
x<12
.
x-4>-8
x>-4
.
,
.
Ed
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
(a)
While the x remains inside the absolute value we will not be able to solve for it. So we must begin to find a way to write equivalent inequalities (yes, it takes two) which do not use absolute value.
Absolute value stands for the distance of a number from zero on a number line, regardless of the direction. So your problem says "The distance of x-4 from zero is between 2 and 8." If we picture a number line and think about this we'll realize, I hope, that either x-4 is between 2 and 8 or it is between -2 and -8:
or
Now that the x is out of the absolute value, we can "get at it" in order to solve for it. The inequalities are called trichotomies because they have three parts: left right and center. We can solve these as trichotomies or we can rewrite them more conventionally. Solving them as trichotomies is quicker but it involves steps that are unusual and, therefore, more difficult to remember and get right. I'll do both:- As trichotomies: Here we have three parts and we must add. subtract, multiply or divide all three parts at the same time in order to solve them correctly. This is the kind unusual steps we use on trichotomies. Normally we are adding numbers to both (2) sides of an equation or inequality. To solve your trichotomies, we add 4 to all three parts:
or
In interval notation this is:
(6, 12) or (-4, 2) - As "normal" inequalities:
- Rewrite the trichotomies as a pair of inequalities. The form of this pair will be:
left < center and center < right
using whatever inequality symbols (less than, less than or equal to, greater than, etc.) you had to begin with. Note how the center portion of the trichotomy is used twice. Use the entire center portion of the trichotomy. Do not try to split it into two parts, one for the first inequality and another for the right inequality. Rewriting our trichotomies we get:
( and ) or ( and )
Since we now have 4 inequalities, I have used parentheses to group each of the pairs. - Solve each inequality. Although we do have 4 inequalities, they are "normal" inequalities and quite simple to solve using normal steps. Solving our set of inequalities we get:
( and ) or ( and
In interval notation this is:
(6, 12) or (-4, 2)
(b)
If we were to multiply this out we would end up with an term (among others). This makes this a 4th degree inequality. To solve inequalities of degree 2 or more, start with the same steps we use on equations of degree 2 or more:- Get one side equal to zero.
- Factor the non-zero side.
Let's get started:
1. One side equal to zero. We already have one side equal to zero. (If the right side had been anything else but zero we would have to multiply out the left side and then subtract the terms on the right from both sides.)
2. Factor. Fortunately the left side is already partially factored. We just need to make sure the left side is factored completely. can be factored as a difference of squares and is a fairly simple trinomial to factor. Now we have:
If this was an equation we would just set each factor equal to zero and solve the 4 equations. But this is not an equation and, unfortunately, it is not correct to set each of these factors greater than zero and then solve. The rest of the solution is a bit tricky, especially if we are unable to order the factors (i.e. list them from lowest to highest). Fortunately we can order our factors (since the x's all have coefficients of 1. Rewriting our inequality in order of lowest factor to highest we have:
Make sure this makes sense to you. No matter what number x is, won't x-4 be a lower number than all the other factors? And won't x+5 be a higher number that the other factors? Etc.
Now why does ordering the factors help?- We have an inequality that says the product of 4 numbers is greater than 0. In other words the product is positive.
- When you multiply 4 numbers, what are the possible ways you can end up with a positive result? Answer: One of the following must be true:
- All four factors are positive
- All four factors are negative
- Two factors are positive and two are negative.
- To solve our inequality we will will need inequalities for each of these possibilities.
- Writing these inequalities is much easier when you know the order of the factors.
So our solution will come from inequalities that say:
(All 4 factors are positive) or (all 4 factors are negative) or (2 of each)
Let's look at each:- All four factors are positive. If the smallest factor is positive, the rest of them (which are larger) would also have to be positive. (Think about this and think about a number line where larger means "to the right of". It should end up making sense.) So the only way our 4 factors will all be positive is if:
- All four factors are negative. If the largest factor is negative, the rest of them (which are smaller) would also have to be negative. (Think about this and think about a number line where smaller means "to the left of". It should end up making sense.) So the only way our 4 factors will all be negative is if:
- Two positive and two negative. Using similar logic as above, we should find that this can happen only if the lower two factors are the negative ones and the larger two factors are the positive ones. And this will happen if and
So
(All 4 factors are positive) or (all 4 factors are negative) or (2 of each)
translates into:
() or () or ( and )
Now we solve these:
() or () or ( and )
In interval notation this is:
(4, ) or (, -5) or {-4, 1)
If we had not been able to order our factors or if we did not know how to take advantage of factors that can be ordered, this would have been much harder. We would have had to rewrite
(All 4 factors are positive) or (all 4 factors are negative) or (2 of each)
with
{4 inequalities saying each factor is positive) or (4 inequalities saying each factor is negative) or (many, many inequalities for each possible combination of factors saying 2 are positive and 2 are negative)
Then we would have to solve all of these inequalities, rejecting the ones that end up being impossible and "condensing" some of the others. With 4 factors this a lot of work.
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