In interval notation the solution set is 2, inf. There is another way to use a graphing utility to solve this inequality. In order to satisfy both inequalities, a number must be in both solution sets. So the numbers that satisfy both inequalities are the values in the intersection of the two solution sets, which is the set -2, 4 in interval notation.
Note: When we solved the two inequalities separately, the steps in the two problems were the same. Therefore, the double inequality notation may be used to solve the inequalities simultaneously. In Example 4 above we were looking for numbers that satisfied both inequalities. Here we want to find the numbers that satisfy either of the inequalities. This corresponds to a union of solution sets instead of an intersection.
To make sense of these statements, think about a number line. The absolute value of a number is the distance the number is from 0 on the number line. This is the set of numbers between -a and a. This means numbers that are either larger than a, or less than -a. Note: -2 and 3 are not in the solution set of the inequality. We are looking for values of x where the polynomial is negative. The solution set of the inequality corresponds to the region where the graph of the polynomial is below the x-axis.
The critical numbers -2 and 3 are the places where the graph intersects the x-axis. The critical numbers divide the x-axis into three intervals called test intervals for the inequality.
We are going to use the fact that polynomial functions are continuous. This means that their graphs do not have any breaks or jumps. Since we have found all the x-intercepts of the graph of x 2 - x - 6, throughout each test interval the graph must be either above the x-axis or below it.
This is where we need to know that the graph does not have any breaks. This means that we may choose any number we like in a test interval and evaluate the polynomial at that number to see if the graph is above or below the x-axis throughout that test interval. When a product of two numbers is equal to 0, then at least one of the numbers must be 0. However, a product of two negative numbers is not negative, so this approach is not useful for solving inequalities.
This problem is much more difficult than the inequality in the previous example! It is not easy to factor, so we will not be able to find the exact values of the critical numbers.
We will use a graphing utility to approximate the critical numbers. The graph of the polynomial is shown below. The critical numbers are approximately In this problem we looking for regions where the graph is above the axis. However, the meaning of this is difficult to visualize—what does it mean to say that an expression , rather than a number, lies between two points? Finally, it is customary though not necessary to write the inequality so that the inequality arrows point to the left i.
However, this is wrong. What numbers work? How about ? By playing with numbers in this way, you should be able to convince yourself that the numbers that work must be somewhere between and This is one way to approach finding the answer. The other way is to think of absolute value as representing distance from 0.
Once again, we conclude that the answer must be between and This answer can be visualized on the number line as shown below, in which all numbers whose absolute value is less than 10 are highlighted. It is not necessary to use both of these methods; use whichever method is easier for you to understand. It is necessary to first isolate the inequality:. Now think about the number line. Therefore, it must be either greater than 8 or less than Expressing this with inequalities, we have:.
We now have 2 separate inequalities. Consider them independently. Now think: the absolute value of the expression is greater than —3. What could the expression be equal to? And 0.
And 7. And — Absolute values are always positive, so the absolute value of anything is greater than —3! All numbers therefore work. Privacy Policy. Skip to main content. Introduction to Equations, Inequalities, and Graphing. Search for:. Learning Objectives Explain what inequalities represent and how they are used. Key Takeaways Key Points An inequality describes a relationship between two different values. Inequalities are particularly useful for solving problems involving minimum or maximum possible values.
Key Terms number line : A visual representation of the set of real numbers as a series of points. Learning Objectives Solve inequalities using the rules for operating on them. If both sides of an inequality are multiplied or divided by the same positive value, the resulting inequality is true.
If both sides are multiplied or divided by the same negative value, the direction of the inequality changes. Inequalities involving variables can be solved to yield all possible values of the variable that make the statement true.
Key Terms inequality : A statement that of two quantities one is specifically less than or greater than another. Learning Objectives Solve a compound inequality by balancing all three components of the inequality. There are two statements in a compound inequality. Learning Objectives Solve inequalities with absolute value.
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