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Question 552098: solve the following inequality for x
x^2-5x+8<4
Please help
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is:
x^2 - 5x + 8 < 4
subtract 4 from both sides of this equation to get:
x^2 - 5x + 4 < 0
the left side of this equation is a quadratic equation that can be factored.
you factor it by setting it equal to 0.
you get:
x^2 - 5x + 4 = 0 which factors out to be:
(x-4)*(x-1) = 0
since (x-4) and (x-1) are factors of x^2 - 5x + 4, then y = x^2 - 5x + 4 becomes equivalent to y = (x-4)*(x-1).
your equation becomes:
(x-4)*(x-1) < 0
you now need to find out when this equation becomes 0.
if (x-4) and (x-1) are both positive, the equation will be positive because a positive times a positive results in a positive.
if (x-4) and (x-1) are both negative, the equation will be positive because a negative times a negative results in a positive.
if they are mixed, i.e.:
(x-4) is positive and (x-1) is negative or:
(x-4) is negative and (x-1) is positive then:
the result will be negative because a positive times a negative results in a negative and a negative times a positive results in a negative.
you need to find the values of x where:
(x-4) is positive and (x-1) is negative, and where:
(x-4) is negative and (x-1) is positive.
let's look at each of these factors.
for (x-4), we get 2 equations:
x-4 > 0 (positive) and x-4 < 0 (negative)
x-4 > 0 when x > 4
x-4 < 0 when x < 4
for (x-1), we get 2 equations:
x-1 > 0 (positive) and x-1 < 0 (negative)
x-1 > 0 when x > 1
x-1 < 0 when x < 1
the result will be negative when:
(x-4) is positive and (x-1) is negative.
this happens when:
x > 4 and x < 1
since this is impossible, we reject this option.
the result will also be negative when:
(x-4) is negative and (x-1) is positive.
this happens when:
x < 4 and x > 1
this can be rewritten a:
1 < x < 4.
that should be your answer.
x^2 - 5x + 8 < 4 when 1 < x < 4.
we can graph the equation of y = x^2 - 5x + 8 to see if we did this right.
that graph of that equation is shown below:
we draw a horizontal line at y = 4 to show us where the demarcation point for the value of y is.
we then draw a vertical line at x = 1 and x = 4 to show us where the demarcation point for the value of x is.
the graph now look like this:
you can see that the value of y is less than 4 when x i greater than 1 and x is less than 4.
not that the original equation is:
x^2 - 5x + 8 < 4
when we subtract 4 from both sides of this equation, we get:
x^2 - 5x + 4 < 0
when we graph the equation of y = x^2 - 5x + 4, we get:
you can see that the value of y is less than 0 when x is greater than 1 and x i less than 4.
by subtracting 4 from both sides of the original equation, we did not change the basic inequality.
it remained the same.
the solution was x is greater than 1 and x is less than 4 either way.
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