When you have to solve applied problems or "word problems" the following stepwise procedure will be very helpful:
- Read the problem carefully - as many times as you need to understand the information and the question.
- Introduce a letter to denote an unknown quantity. Usually, this quantity is what you have to evaluate.
- Write a formula or equation that connects the quantities appearing in the given information.
- Draw a figure if possible.
- Set up a table or a list of values of the quantities given in the problem.
- Set up an equation, using the table or list of values.
- Solve the equation.
- Check that your solution is consistent with the given information.
CLICK HERE for an example of solution of an applied problem using this procedure.
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When you have to solve an inequality involving the absolute value of an expression, you may replace this inequality by TWO inequalities without the absolute value. The two replacement inequalities will depend on how the initial inequality has been set up. Here, we consider the two different cases using a linear expression.
CASE 1
When the initial inequality is of the form |ax+b| < c we replace it by -c < ax+b < c .
Note that there are two inequalities here and the way they are written implies that both first AND the second inequalities must be satisfied. The solution set will then be an INTERSECTION of two sets.
CASE 2
When the initial inequality is of the form |ax+b| > c we replace it by ax+b < -c OR ax+b > c .
Again, we have two inequalities here and the way they are written implies that the first OR the second inequality must be satisfied. The solution set will then be the UNION of two sets.
CLICK HERE for examples of solution of these types of absolute value inequalities.
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An effective stepwise procedure for solving rational inequalities is the following:
- Move all the terms to one side (usually the left side) of the inequality with 0 on the other side.
- Combine the non-zero terms into a single rational expression.
- Factor the numerator and denominator of this expression.
- Find the critical points. That is, the points where the numerator or denominator is zero.
- Draw the real line and divide it into intervals separated by the critical points.
- Set up a sign diagram for the intervals.
- Select the intervals that satisfy the requirement of the inequality.
- If the problem also involves an equation, select the appropriate critical points that satisfy the equation. Usually, these are the values of x which make the numerator equal to zero.
- Display your answer in the form of a set (or union of sets), an interval (or union of intervals) or a graph on the real line.
Warning: Remember that you should not multiply both sides of an inequality by an expression that contains the unknown and can change sign.
CLICK HERE for an example of the solution of a rational inequality using this procedure.
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We consider a rational function with equation y = f(x), where f(x) = P(x)/Q(x). Here, P(x) and Q(x) are polynomials of degrees m and n, respectively. We assume that P and Q do not have any common factors.
To graph the function we may use the following stepwise procedure:
- Find the domain of f.
- Find the critical points of f. That is, the values of x where P(x)=0 or Q(x)=0.
- The x-intercepts are the values of x where P(x)=0.
- Find the y-intercept.
- The vertical asymptotes are the vertical lines with values of x for which Q(x)=0.
- Draw the real line and divide it into intervals separated by the critical points.
- Set up a sign diagram for the intervals.
- Near the vertical asymptotes, the graph goes up towards plus infinity in intervals where f is positive and goes down towards minus infinity in intervals where f is negative.
- Determine the behavior of y as x approaches plus or minus infinity. The following rules may be used for this:
- If m is less than n then y = 0 is a horizontal asymptote.
- If m = n then y = k is a horizontal asymptote, where k is obtained by dividing the leading coefficient of P by the leading coefficient of Q.
- If m = n+1 then there is an oblique asymptote and its equation must be obtained by dividing P by Q.
- If m is larger than n+1 then there is no linear asymptote.
- Compute the values of y for a few conveniently chosen values of x and plot the corresponding points.
- Complete the graph.
For an example of the graph of a rational function click HERE.
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The standard form of the equation, the graph and important features of each of the following conic sections are given in graphics format.
