Question 1200141
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As the other tutor indicates, the method for solving all three of your problems is the same, so I will only address the first.<br>
The response from that other tutor shows a standard solution method taught in virtually all references.  With that method, we find the x values where the function value is 0 and divide the x axis into intervals using those x values; then we use a test point in each interval to find the interval(s) in which the given inequality is satisfied.<br>
For a quadratic inequality, with two zeros creating 3 intervals, that process is relatively efficient.  But for higher degree polynomials, or for rational functions, where the number of intervals might be much greater than 3, there is a more efficient way to determine the correct intervals.<br>
This more efficient way for finding the correct intervals uses the fact that, as we "walk" along the x axis, the sign of the function can change only at the zeros of the function (or, in the case of rational functions, the zeros of either the numerator or denominator).<br>
It is usually easiest to start to the right of every zero, where all factors of the function are positive, and move to the left to see where the sign of the function changes.<br>
For your first example....<br>
{{{x^2-5x+4>0}}}
{{{(x-1)(x-4)>0}}}<br>
The zeros are at x=1 and x=4.<br>
So pick a value greater than 4 and see that the function value is positive.<br>
Then, as you "walk" along the number line to the left, the function value changes sign each time you pass one of the roots.  So the function value is negative between 1 and 4, and positive again for x less than 1.<br>
ANSWER: The inequality is satisfied on (-infinity,1) and (4,infinity).<br>
Of course, a completely different solution method, once the roots x=1 and x=4 are determined, is to know that the graph is an upward-opening parabola, so the inequality is NOT satisfied only between the two roots.<br>