Problem 5 - Intermediate Algebra

Question

Fill in the blanks. a. The solution set of a compound inequality containing the word and includes all numbers that make ______ inequalities true. b. The solution set of a compound inequality containing the word or includes all numbers that make ______ , or the other, or ______ inequalities true.

Step-by-Step Solution

Answer

a. both; b. one, both

Step 1: Understanding 'and' in Compound Inequalities

When dealing with compound inequalities connected by the word **and**, it implies that both conditions in the inequalities must be satisfied. Therefore, the solution set is the intersection of the solution sets of the individual inequalities; this includes numbers that satisfy both inequalities simultaneously.

Step 2: Clarification for Blank (a)

Since the word **and** indicates that both inequalities need to be true at the same time, the correct word to fill in the blank is **both**.

Step 3: Understanding 'or' in Compound Inequalities

For compound inequalities connected by the word **or**, the condition is less strict: a number satisfies the compound inequality if it meets either one of the conditions or both. This means the solution set is the union of the solution sets of the individual inequalities.

Step 4: Clarification for Blank (b)

The word **or** implies flexibility as satisfying either inequality is sufficient. Therefore, the first blank is filled with **one** and the second with **both**, meaning a number can satisfy one inequality, the other, or both to be part of the solution set.

Key Concepts

intersection of solution sets

The concept of the "intersection of solution sets" is essential in understanding compound inequalities connected by the word "and." In mathematics, the intersection of two sets refers to the elements that are common to both sets.

In the context of compound inequalities, when we say that two inequalities are connected by "and," it means that the solution set will consist only of the numbers that satisfy both inequalities at the same time. To find the intersection, imagine overlapping two sets and identifying the shared region.

• If the solution to inequality A produces the set {x | x > 2},
• And the solution to inequality B is {x | x < 5},
• Then the intersection is {x | 2 < x < 5}.

This set includes only those values of \( x \) which satisfy both conditions. Understanding this concept helps in accurately solving and graphing compound inequalities that require both conditions to be true.

union of solution sets

The "union of solution sets" is a concept applied to compound inequalities containing the word "or." When dealing with the word "or," the requirement is more inclusive compared to "and."

The solution set for inequalities joined by "or" includes any number that satisfies at least one of the inequalities. This means that if a number works for the first inequality or the second, or even both, it is part of the solution set.

For example:

• Suppose inequality C has solutions {x | x > 5},
• And inequality D provides {x | x < 3},
• The union would be {x | x > 5 or x < 3}.

In this instance, you accept any number greater than 5 or less than 3. No middle ground or overlap needed.
Understanding the union helps in recognizing the broader range of solutions and simplifies the method to include all possibilities.

inequality solutions

"Inequality solutions" involve finding the set of numbers that satisfy a given inequality, defining a range of values rather than a fixed number. Solving inequalities is similar to solving equations, but there are vital rules to follow.

• When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
• Adding or subtracting the same value from both sides keeps the inequality sign the same.

In solving either simple or compound inequalities, the goal is to isolate the variable. Simple inequalities like \( x + 3 > 5 \) lead to solutions such as \( x > 2 \). For compound inequalities, you may deal with more than one inequality simultaneously, using "and" or "or" to combine conditions, which leads to intersection or union of sets.
The graphical representation of these solutions often assists in visualizing the solution set, making the inequality solutions an integral part of algebra and calculus.