![\"\"](\"/images/stars/stars-13.gif\")
You can put this solution on YOUR website!
your equation is:
\n" ); document.write( "x^2 - 5x + 8 < 4
\n" ); document.write( "subtract 4 from both sides of this equation to get:
\n" ); document.write( "x^2 - 5x + 4 < 0
\n" ); document.write( "the left side of this equation is a quadratic equation that can be factored.
\n" ); document.write( "you factor it by setting it equal to 0.
\n" ); document.write( "you get:
\n" ); document.write( "x^2 - 5x + 4 = 0 which factors out to be:
\n" ); document.write( "(x-4)*(x-1) = 0
\n" ); document.write( "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).
\n" ); document.write( "your equation becomes:
\n" ); document.write( "(x-4)*(x-1) < 0
\n" ); document.write( "you now need to find out when this equation becomes 0.
\n" ); document.write( "if (x-4) and (x-1) are both positive, the equation will be positive because a positive times a positive results in a positive.
\n" ); document.write( "if (x-4) and (x-1) are both negative, the equation will be positive because a negative times a negative results in a positive.
\n" ); document.write( "if they are mixed, i.e.:
\n" ); document.write( "(x-4) is positive and (x-1) is negative or:
\n" ); document.write( "(x-4) is negative and (x-1) is positive then:
\n" ); document.write( "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.
\n" ); document.write( "you need to find the values of x where:
\n" ); document.write( "(x-4) is positive and (x-1) is negative, and where:
\n" ); document.write( "(x-4) is negative and (x-1) is positive.
\n" ); document.write( "let's look at each of these factors.
\n" ); document.write( "for (x-4), we get 2 equations:
\n" ); document.write( "x-4 > 0 (positive) and x-4 < 0 (negative)
\n" ); document.write( "x-4 > 0 when x > 4
\n" ); document.write( "x-4 < 0 when x < 4
\n" ); document.write( "for (x-1), we get 2 equations:
\n" ); document.write( "x-1 > 0 (positive) and x-1 < 0 (negative)
\n" ); document.write( "x-1 > 0 when x > 1
\n" ); document.write( "x-1 < 0 when x < 1
\n" ); document.write( "the result will be negative when:
\n" ); document.write( "(x-4) is positive and (x-1) is negative.
\n" ); document.write( "this happens when:
\n" ); document.write( "x > 4 and x < 1
\n" ); document.write( "since this is impossible, we reject this option.
\n" ); document.write( "the result will also be negative when:
\n" ); document.write( "(x-4) is negative and (x-1) is positive.
\n" ); document.write( "this happens when:
\n" ); document.write( "x < 4 and x > 1
\n" ); document.write( "this can be rewritten a:
\n" ); document.write( "1 < x < 4.
\n" ); document.write( "that should be your answer.
\n" ); document.write( "x^2 - 5x + 8 < 4 when 1 < x < 4.
\n" ); document.write( "we can graph the equation of y = x^2 - 5x + 8 to see if we did this right.
\n" ); document.write( "that graph of that equation is shown below:
\n" ); document.write( "
\n" ); document.write( "we draw a horizontal line at y = 4 to show us where the demarcation point for the value of y is.
\n" ); document.write( "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.
\n" ); document.write( "the graph now look like this:
\n" ); document.write( "
\n" ); document.write( "you can see that the value of y is less than 4 when x i greater than 1 and x is less than 4.
\n" ); document.write( "not that the original equation is:
\n" ); document.write( "x^2 - 5x + 8 < 4
\n" ); document.write( "when we subtract 4 from both sides of this equation, we get:
\n" ); document.write( "x^2 - 5x + 4 < 0
\n" ); document.write( "when we graph the equation of y = x^2 - 5x + 4, we get:
\n" ); document.write( "
\n" ); document.write( "you can see that the value of y is less than 0 when x is greater than 1 and x i less than 4.
\n" ); document.write( "by subtracting 4 from both sides of the original equation, we did not change the basic inequality.
\n" ); document.write( "it remained the same.
\n" ); document.write( "the solution was x is greater than 1 and x is less than 4 either way.
\n" ); document.write( " \n" ); document.write( "