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Solving Inequalities: An Overview (page 3 of 3)

Sections: Linear inequalities, Quadratic inequalities, Other inequalities


General Polynomial Inequalities

  • Solve x5 + 3x4 23x3 51x2 + 94x + 120 > 0

    First, I factor to find the zeroes:

      x5 + 3x4 23x3 51x2 + 94x + 120

        = (x + 5)(x + 3)(x + 1)(x 2)(x 4) = 0

    ...so x = 5, 3, 1, 2, and 4 are the zeroes of this polynomial. (Review how to solve polynomials, if you're not sure how to get this solution.)

    To solve by the Test-Point Method, I would pick a sample point in each interval, the intervals being (negative infinity, 5), (5, 3), (3, 1), (1, 2), (2, 4), and (4, positive infinity). As you can see, if your polynomial or rational function has many factors, the Test-Point Method can become quite time-consuming.

    To solve by the Factor Method, I would solve each factor for its positivity: x + 5 > 0 for x > 5; x + 3 > 0 for x > 3; x + 1 > 0 for x > 1; x 2 > 0 for x > 2; and x 4 > 0 for x > 4. Then I draw the grid:

      blank factor table

    ...and fill it in:

      factor table with signs

    ...and solve:

      factor table with signs

    Then the solution (remembering to include the endpoints, because this is an "or equal to" inequality) is the set of x-values in the intervals [5, 3], [1, 2], and [4, positive infinity]

As you can see, if your polynomial or rational function has many factors, the Factor Method can be much faster.


Rational Inequalities

 

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  • Solve x/(x 3) < 2

    First off, I have to remember that I can't begin solving until I have the inequality in "= 0" format.

      x/(x - 3) - 2 <= 0

    Now I need to convert to a common denominator:

      x/(x - 3) - 2(x - 3)/(x - 3) <= 0

    ...and then I can simplify:   Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

      (-x + 6)/(x - 3) <= 0

    The two factors are x + 6 and x 3. Note that x cannot equal 3, or else I would be dividing by zero, which is not allowed. The first factor, x + 6, equals zero when x = 6. The other factor, x 3, equals zero when x = 3. Now, x cannot actually equal 3, so this endpoint will not be included in any solution interval (even though this is an "or equal to" inequality), but I need the value in order to figure out what my intervals are. In this case, my intervals are (negative infinity, 3), (3, 6], and [6, positive infinity). Note the use of brackets to indicate that 6 can be included in the solution, but that 3 cannot.

    Using the Test-Point Method, I would pick a point in each interval and test for the sign on the result. I could use, say, x = 0, x = 4, and x = 7.

    Using the Factor Method, I solve each factor: x + 6 > 0 for x > 6, or x < 6; x 3 > 0 for x > 3. Then I do the grid:

      blank sign chart

    ...fill in the signs on the factors:

      factor signs filled in

    ...and solve for the sign on the rational function:

      interval signs filled in

    So the solution is all x's in the intervals (negative infinity, 3) and [6, positive infinity).


There is another way to solve inequalities. You still have to find the zeroes (x-intercepts) first, but then you graph the function, and just look: wherever the graph is above the x-axis, the function is positive; wherever it is below the axis, the function is negative. For instance, for the first quadratic exercise, y = x2 3x + 2 > 0, we found the zeroes at x = 1 and x = 2. Now look at the graph:

    graph of y = x^2 – 3x + 2

On the graph, the solution is obvious: you would take the two intervals (but not the interval endpoints) where the line is above the x-axis.

Or that huge polynomial we did at the top of this page: x5 + 3x4 23x3 51x2 + 94x + 120 > 0. We found the zeroes of the polynomial, being x = 5, x = 3, x = 1, x = 2, and x = 4. Now look at the graph:

    graph of y = x^5 + 3x^4 – 23x^3 – 51x^2 + 94x + 120

On the picture, the solution is obvious: take the three intervals (together with the interval endpoints) where the line is above the x-axis.

As you can probably guess, a graphing calculator can save you a lot of time on these inequalities if you understand what you're doing. You should still show your work and reasoning, but don't be shy about using the pictures to confirm the algebra.

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Cite this article as:

Stapel, Elizabeth. "Solving Inequalities: An Overview." Purplemath. Available from
    http://www.purplemath.com/modules/ineqsolv3.htm. Accessed
 

 



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