Does changing the compound inequality $x>-3$ and $x<3$ from "and" to "or" change the solution set? Explain.
Answer (2)
The compound inequality -3"> x > − 3 and x < 3 means x must be greater than − 3 AND less than 3 , which is − 3 < x < 3 .
The compound inequality -3"> x > − 3 or x < 3 means x must be greater than − 3 OR less than 3 , which is all real numbers.
The solution sets are different, so changing "and" to "or" changes the solution set.
"And" requires both conditions to be true, while "or" requires at least one condition to be true, leading to different solution sets. The final answer is Yes, the solution set changes.
Explanation
Understanding the Problem We are asked to determine if changing the 'and' in the compound inequality -3"> x > − 3 and x < 3 to 'or' changes the solution set and explain why.
Solving the 'and' Inequality First, let's solve the compound inequality -3"> x > − 3 and x < 3 . This means we are looking for all values of x that satisfy both inequalities simultaneously. In other words, x must be greater than − 3 AND less than 3 . This can be written as − 3 < x < 3 .
Solving the 'or' Inequality Next, let's solve the compound inequality -3"> x > − 3 or x < 3 . This means we are looking for all values of x that satisfy either inequality. In other words, x must be greater than − 3 OR less than 3 . Let's consider the number line. The inequality -3"> x > − 3 includes all numbers to the right of − 3 . The inequality x < 3 includes all numbers to the left of 3 . The union of these two sets includes all real numbers except for the interval from 3 to − 3 including 3 and − 3 . However, since x can be any real number, and any real number is either greater than − 3 or less than 3 , the solution set is all real numbers.
Comparing the Solution Sets Now, let's compare the solution sets. The solution to -3"> x > − 3 and x < 3 is − 3 < x < 3 . The solution to -3"> x > − 3 or x < 3 is all real numbers. Since these solution sets are different, changing 'and' to 'or' does change the solution set.
Explaining the Difference The word 'and' implies that both conditions must be true simultaneously. The word 'or' implies that at least one of the conditions must be true. In this case, when we use 'and', we are looking for the intersection of the two solution sets. When we use 'or', we are looking for the union of the two solution sets.
Final Answer Therefore, changing the compound inequality -3"> x > − 3 and x < 3 from "and" to "or" changes the solution set.
Examples
In electrical engineering, consider a circuit that requires a voltage to be both greater than -3 volts AND less than 3 volts to operate correctly. If the condition changes to the voltage being greater than -3 volts OR less than 3 volts, the circuit will operate under almost any voltage condition, which is a completely different scenario. This illustrates how changing 'and' to 'or' can drastically alter the conditions under which a system operates.
Changing the compound inequality from 'and' to 'or' changes the solution set. The 'and' inequality − 3 < x < 3 restricts x to a specific range, while the 'or' inequality allows for all real numbers. Therefore, the two inequalities represent different solution sets.
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