solve inequality (x - 4)(x - 5)(x - 7) < 0?

Question

please help

Answer 1

For (x - 4)(x - 5)(x - 7) to be negative there are at most four possibilites. Any one of the three terms must be negative, or all three must be negative. Lets look at each case in turn.

x-4<0, x-5>0 & x-7>0
x<4 AND x>5 & x>7
x cant be less than 4 AND greater than 5, so this is impossible.

x-5<0, x-4>0 & x-7>0
x<5 AND x>4 AND x>7
Similarly, x cant be less than 5 AND greater than 7, so this is impossible

x-7<0, x-4>0 & x-5>0
x<7 AND x>4 AND x>5
This is possible. Any x greater that 5 AND smaller than 7 will work.

Last case
x-4<0 & x-5<0 & x-7<0
x<4 AND x<5 AND x<7
This is possible too. X just has to be less than 4 for all these to be true.

So for the inequality to hold either x<4 OR 5

Answer 2

x<4 or 7>x>5

Answer 3

x<4 or x<5 or x<7

Answer 4

x<4 OR 5

Answer 5

(x - 4)(x - 5)(x - 7) < 0
For the product of the 3 factors to be less than zero
then all 3 factors are less than zero or 1 factor is less than 0 and the other 2 factors are greater than 0
So there are 4 possible ways this can happen

case 1
x - 4 < 0, x - 5 < 0, and x - 7 < 0
x < 4, x < 5, and x < 7
The solution of case 1 is x < 4

case 2
x - 4 < 0, x - 5 > 0, and x - 7 > 0
x < 4, x > 5, and x > 7
The solution of case 2 is no solution

case 3
x - 4 > 0, x - 5 < 0, and x - 7 > 0
x > 4, x < 5, and x > 7
The solution of case 3 is no solution

case 4
x - 4 > 0, x - 5 > 0, and x - 7 < 0
x > 4, x > 5, and x < 7
The solution of case 4 is 5 < x < 7

So the complete solution is
x < 4 or 5 < x < 7

You can verify or do this problem with a graphing calculator
Graph
y = (x - 4)(x - 5)(x - 7)
and see what values of x is the graph below the x-axis (y < 0)

Answer 6

Check when the number of negative factors is odd.

If x<4:
x - 4 < 0
x - 5 < 0
x - 7 < 0
(x - 4)(x - 5)(x - 7) is a product of 3 negative numbers, thus the product is negative.

If 4 < x < 5:
x - 4 > 0
x - 5 < 0
x - 7 < 0
(x - 4)(x - 5)(x - 7) is a product of 3 numbers out of which 2 are negative, thus (x - 4)(x - 5)(x - 7) > 0

If 5 < x < 7:
x - 4 > 0
x - 5 > 0
x - 7 < 0
(x - 4)(x - 5)(x - 7) is a product of 2 positive numbers and 1 negative, thus (x - 4)(x - 5)(x - 7) < 0

If x > 7:
x - 4 > 0
x - 5 > 0
x - 7 > 0

Thus, (x - 4)(x - 5)(x - 7) > 0

Take x<4 or 5 < x < 7

Answer 7

Here's a method that my maths lecturer taught us:

1) Once you have completely factorised a polynomial, draw a number line. Here are the instructions on how to make one:

2) Draw a moderately long horizontal arrow pointing to the right.

3) On the top of the arrowhead, write the polynomial in the question, in this case, you write (x-4)(x-5)(x-7)

4) On the bottom of the arrowhead, you write the letter x.

5) Now ask yourself this: What values of x will make this polynomial equal to 0? The values you will get are 4, 5, and 7.

6) On the line, put 3 markings are below these markings, write the numbers 4, 5 and 7 in ascending order.

7) On top of these markings, write the number 0. This will tell you that for these values of x, the polynomial is equal to zero.

8) Now, look at your polynomial. All of the factors of the polynomial have been reduced to the degree 1. You must take note of this.

9) Now, try obtaining the coefficient of the highest power of x in the polynomial. The coefficient in positive, right?

10) Now we can deduce whether the answers given in the polynomial will be negative or positive. Since the coefficient is positive, write the "+" sign on the interval, between the last zero and the arrowhead.

11) Since the factor from which you obtained x = 7 is reduced to power 1, we write the "-" sign in the preceding interval, which is between the 0 from x = 5 and the 0 from x = 7.

12) Continuing to do this, your final number line should be something like this:

http://i20.photobucket.com/albums/b202/SimpleRobin/numberline.jpg

13) Now we can analyse. Looking back at your question, you are looking for sets of values of x such that (x-4)(x-5)(x-7) is LESS THAN 0. So, looking at the number line, we find that there are negative signs in two places (x < 4 and 5
Therefore , these are your answers:

x < 4 OR 5 < x < 7 (Ans)

Answer 8

A good way to solve inequalities with products is to solve by cases.
there are four critical values 4, 5 and 7.
Case one:
if x<4 all three factors are negative, so the product is negative. this will be part of the solution.
Case 2:
4 Case 3:
5 Finally Case 4: If x>7 has all three factors positive so that will not work.
You final answer is from case 1 & 3
x < 4 ∪ 5