How to Graph a Compound Inequality on the Number LineStep 1: Graph each individual inequality of the compound inequality by shading the values on the number line that satisfy that inequality. If the inequality symbol is {eq}\ge {/eq} or {eq}\le {/eq}, make a filled-in circle at the endpoint of the graph to indicate that that value is included; otherwise, make an open circle at the endpoint to indicate that that value is not included. The end of the graph that goes on forever should have an arrow indicating that fact. Show
Step 2: If the compound inequality includes the word AND, the graph of the compound inequality is the intersection of the two graphs from Step 1. Only shade the section of the number line that is part of both graphs. If the compound inequality includes the word OR, the graph of the compound inequality is the union of the two graphs from Step 1. Include both of these graphs in the final graph. How to Graph a Compound Inequality on the Number Line: VocabularyCompound Inequality: A statement that combines two inequalities with AND or OR is a compound inequality. For a value to satisfy a compound inequality that includes the word AND, the value must satisfy both inequality statements. For a value to satisfy a compound inequality including the word OR, the value need only satisfy one of the inequality statements. Now let's practice graphing two compound inequalities, one with AND and one with OR. How to Graph a Compound Inequality on the Number Line: Example with ANDGraph the compound inequality {eq}x>-2\text{ and }x\le 5. {/eq} Step 1: First we graph each of the individual inequalities. We don't include the point {eq}-2 {/eq} in the graph of {eq}x>-2 {/eq}, and the graph goes on forever to the right from this point. We do include the point {eq}5 {/eq} in the graph of {eq}x\le 5 {/eq}, and the graph goes on forever to the left from this point. Step 2: Because we want to graph the values of {eq}x {/eq} that satisfy both inequalities, we only shade the part of the number line that is part of both graphs: How to Graph a Compound Inequality on the Number Line: Example with ORGraph the compound inequality {eq}x\le -3 \text{ or }x>1. {/eq} Step 1: Because the graph of this compound inequality is the union of two inequalities, we can graph the two inequalities on the same number line to get our final graph. We must include the endpoint {eq}-3 {/eq} and not include the endpoint {eq}1 {/eq}. Step 2: Here is the final graph: Get access to thousands of practice questions and explanations! A compound inequality is a combination of two inequalities that are combined by either using "and" or "or". The process of solving each of the inequalities in the compound inequalities is as same as that of a normal inequality but just while combining the solutions of both inequalities depends upon whether they are clubbed by using "and" or "or". Let us see the process of solving compound inequalities and also how to graph them on the number line.
What is a Compound Inequality?A compound inequality is an inequality that merges two inequalities either by using "and" or "or". Sometimes "and" won't be specifically mentioned but it is understood. For example, 1 < x < 3 is nothing but "x > 1 and x < 3". On the other hand, a compound inequality with "or" is always specifically mentioned using "or". The two types of compound inequalities are conjunction and disjunction and the following are the differences between them,
Compound Inequality GraphWe can graph the compound inequalities on the number line. Keep the following things in mind when graphing compound inequalities.
Example of Graphing Compound Inequality with ANDGraph the compound inequality x > 1 AND x ≤ 4. We will graph each inequality on the number line as follows:
Finally, we look for the values on the number line that satisfy "BOTH" inequalities as the given compound inequality has "AND" in it. It is very clear from both arrows above that the region of intersection (common region) of both arrows is from 1 to 4 where 1 is not included and 4 is included. i.e., the solution of the given compound inequality is 1 < x ≤ 4 and is shown by a "orange" line on the number line below. In the interval notation, the solution can be written as (1, 4]. Example of Graphing Compound Inequality with ORGraph the compound inequality x ≤ -1 OR x > 2. We will graph each inequality on the number line as follows:
Since the given inequality has an "OR" in it, we just use "union" between them in the solution. Hence the solution is x ≤ -1 OR x > 2 itself. In the interval notation, the solution is, (-∞, -1] ∪ (2, ∞). The solution is shown with "orange" line on the number line below. Solving Compound InequalitiesWe know that a compound inequality is the merging of two simple inequalities. Hence, each of them can be solved in the same way as we solve a simple inequality (to know how to solve a simple inequality, click here) and then we club them according to what they gave (AND / OR) just like how it was explained in the previous section. Here is the step-by-step explanation of solving compound inequalities.
If we are able to analyze the solution without graphing, then we can avoid Step 3. Graphing compound inequalities is not compulsory unless it is specifically mentioned to graph it. The above procedure is explained with examples (one of each of "AND" and "OR") below. Example of Solving Compound Inequality with ANDSolve the compound inequality -2 < 2x - 3 < 5. We can solve it in two ways: one is without splitting into two; and the other is by splitting into two. Let us see both ways. Method 1: The given compound inequality is -2 < 2x - 3 < 5. Adding 3 on all the sides, -2 + 3 < 2x - 3 + 3 < 5 + 3 1 < 2x < 8 Dividing all the sides by 2, 1/2 < 2x/2 < 8/2 1/2 < x < 4. Hence, the direct solution is 1/2 < x < 4 (or) the interval (1/2, 4). Method 2: Split the given inequality into two inequalities. Then we get 2x - 3 > -2 AND 2x - 3 < 5 Adding 3 on both sides of each inequality, 2x > 1 AND 2x < 8 Dividing each inequality both sides by 2, x > 1/2 AND x < 4 Again, merging them back, 1/2 < x < 4. Hence, the solution is 1/2 < x < 4 (or) the interval (1/2, 4). Example of Solving Compound Inequality with ORSolve the compound inequality -3x - 1 > -7 OR -5x + 2 < -12. Add 1 on both sides of the first inequality and subtract 2 from both sides of the second inequality. -3x > -6 OR -5x < -14 Divide the first inequality on both sides by -3 and the second inequality by -5. Note that the sign of the inequality changes when we divide by a negative number. x < 2 OR x > 2.8 Hence, the solution is (-∞, 2) U (2.8, ∞). Important Notes on Compound Inequalities:
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FAQs on Compound InequalityWhat are Compound Inequalities?Compound inequalities are combinations of two inequalities that are combined by either using "and" or "or". For example, 1 < x < 3 is nothing but "x > 1 and x < 3". What is Compound Inequality Example?A compound inequality, as its name suggests, contains two inequalities with an "AND" or "OR" in between them. For example, -2 < x OR x > 5 is a compound inequality. How to Solve Compound Inequalities?To solve compound inequalities, just solve each of the two inequalities in it separately and combine them using
How to Tell if a Compound Inequality is AND or OR?For compound inequalities, "AND" or "OR" is usually mentioned explicitly like "x < 2 AND x > -1". But sometimes there can be inequality of the form -2 < x < 1 and it obviously means x > -2 AND x < 1. What is Compound Inequality Graph?The compound inequality graph is the graph that shows the solution of the compound inequality. To draw the compound inequality graph:
What is the Process of Graphing Compound Inequalities?For graphing compound inequalities,
What Does And/Or Mean in Compound Inequality?"AND" means each value in the solution interval should satisfy BOTH inequalities. "OR" means each value in the solution interval can satisfy one/both inequalities. What are the Applications of Compound Inequality?The rate of burning of fuel in a two-wheeler is an example of a compound inequality because the fuel burns more if the vehicle is moving either too slow OR too fast. Maximizing profit with respect to two given constraints (linear programming problem) is an example of a compound inequality with "AND". |