In general, graphs of rational functionsĭo have breaks. Use a graphing utility to check your solution.Ī rational expression is one of the form polynomial divided by polynomial.
In this problem we looking for regions where The critical numbers are approximately -2.35, -1.25, and 1.05. The graph of the polynomial is shown below. We will use a graphing utility to approximate We will not be able to find the exact values of the critical numbers. This problem is much more difficult than the inequality in the previous example! It is not easy to factor, so Since we are looking for regions where the graph is below the axis, the solution set is -2 0. Is above the x-axis on the entire interval. 0 2 - 0 - 6 = -6 0, so the graph of y = x 2 - x - 6 x - 6 is above the x-axis on the entire interval (-inf, -2). Return to Contents Polynomial Inequalities That are either larger than a, or less than -a. So the inequality |x| a is satisfied by numbers whose distance from 0 is larger than a. The absolute value of a number is the distance To make sense of these statements, think about a number line. Inequalities involving absolute values can be rewritten as combinations of inequalities. Return to Contents Inequalities Involving Absolute Values The solution set is the union of the two intervals (-inf, -1) and (2, inf). This corresponds to a union of solution sets instead of an intersection.ĭo not use the double inequality notation in this situation. In Example 4 above we were looking for numbers that satisfied both inequalities. The graph of y = 5 - 2x is between the graphs of y = -3 and y = 9. In terms of graphs, this problem corresponds to finding the values of x such that the corresponding point on The problem above is usually written as a double inequality. Inequalities are the values in the intersection of the two solution sets, which is the set (-2, 4) in interval In order to satisfy both inequalities, a number must be in both solution sets.
Return to Contents Combinations of Inequalities The picture below shows the graph of (7-2*x)元 as drawn by the Grapher. (7-2*x)元 has the value 1 for numbers x that satisfy the inequality, and the value 0 for other numbers There is another way to use a graphing utility to solve this inequality. On the graph of y = 7 - 2x is below the point on the graph of y = 3. So we are looking for numbers x such that the point
To satisfy the inequality, 7 - 2x needs to be less than 3. Look at the graphs of the functions on either side of the inequality. Note: When we divided both sides of the inequality by -2 we changed the direction of the inequality. Note: In general we may not multiply or divide both sides of an inequality by an expression with a variable,īecause some values of the variable may make the expression positive and some may make it negative. So, when we multiply the original inequality by -2, we must reverse theĭirection to obtain another true statement. Note that Htmlĭoes not support the standard symbols for "less than or equal to" and "greater than or equal to", 225 #11, 12, 13, 14, 16, 28, 33, 35, 38, 41, 53, 56, 62, 63, 68, 69 Linear Inequalities Combinations of Inequalities Inequalities Involving Absolute Values Polynomial Inequalities Rational InequalitiesĪn inequality is a comparison of expressions by either "less than" (), or "greater than or equal to" (>=). Contents: This page corresponds to § 2.5 (p.
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