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Which Graph Represents the Solution Set of the Compound Inequality? −5
Subtracting 1 from both sides, we have:
1 – 1 < a + 1 - 1
0 < a
Inequality 2:
Subtracting 1 from both sides, we have:
a + 1 – 1 < 8 - 1
a < 7
Now that we have the individual inequalities, we can represent them on a number line and find the overlapping region.
On the number line, mark a point at 0 and shade everything to the right of it to represent the inequality 0 < a. Similarly, mark a point at 7 and shade everything to the left of it to represent the inequality a < 7. The overlapping shaded region between 0 and 7 represents the solution set for the compound inequality −5 < a − 6 < 2.
FAQs:
Q: How do compound inequalities differ from simple inequalities?
A: Simple inequalities involve a single inequality symbol (<, >, ≤, ≥) and express a range of values that satisfy a given condition. Compound inequalities, on the other hand, involve two or more inequalities connected by “and” or “or” operators. They express combined conditions that must be satisfied simultaneously or independently.
Q: What is the meaning of “and” and “or” in compound inequalities?
A: In compound inequalities, “and” is used when both inequalities must be true simultaneously, while “or” is used when either of the inequalities can be true independently. For example, in the compound inequality 1 < x < 5, "and" indicates that x must be greater than 1 and less than 5, while "or" would represent a range where x can be greater than 1 or less than 5.
Q: How do I represent compound inequalities on a graph?
A: To graph compound inequalities, start by solving each inequality separately. Then, represent each inequality on a number line, shading the appropriate regions. The overlapping shaded region represents the solution set of the compound inequality.
Q: Can a compound inequality have no solution?
A: Yes, a compound inequality can have no solution. This occurs when the solution sets of the individual inequalities do not overlap on the number line. In other words, there is no common range of values that satisfies both inequalities simultaneously.
Q: Are there any other methods to solve compound inequalities?
A: Yes, besides graphing, compound inequalities can also be solved algebraically by manipulating the inequalities and isolating the variable. However, graphing is often a visual and intuitive method that provides a clear representation of the solution set.
In conclusion, the graph that represents the solution set of the compound inequality −5 < a − 6 < 2 is a shaded region between 0 and 7 on a number line. By understanding the basics of compound inequalities and their graphical representation, you can effectively solve and interpret such mathematical expressions. [ad_2]