An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Solve the inequality 2 x + 4 1 < x , which gives x < − 4 1 .
The second inequality is x < 9 .
Combine the two inequalities to find the more restrictive condition: x < − 4 1 .
The solution set is x < − 4 1 , so the elements of the set B are all real numbers x such that x < − 4 1 . x < − 4 1
Explanation
Understanding the Problem We are given the set B = { 2 x + 4 1 < x < 9 } . We need to find the values of x that satisfy the compound inequality 2 x + 4 1 < x < 9 . This means we need to solve two inequalities: 2 x + 4 1 < x and x < 9 .
Solving the First Inequality First, let's solve the inequality 2 x + 4 1 < x . We want to isolate x on one side of the inequality. Subtract 2 x from both sides: 4 1 < x − 2 x 4 1 < − x Now, multiply both sides by − 1 . Remember that when we multiply or divide an inequality by a negative number, we need to reverse the inequality sign: x"> − 4 1 > x So, x < − 4 1 .
Solving the Second Inequality Next, we have the inequality x < 9 . This inequality is already solved for x .
Combining the Inequalities Now, we need to combine the two inequalities: x < − 4 1 and x < 9 . Since x must be less than both − 4 1 and 9 , we take the more restrictive condition, which is x < − 4 1 . Therefore, the solution set is all x such that x < − 4 1 .
Final Answer The elements of the set B are all real numbers x such that x < − 4 1 . In interval notation, this is the interval ( − ∞ , − 4 1 ) .
Examples
Consider a scenario where you are designing a simple electrical circuit. The voltage x in the circuit must satisfy the condition 2 x + 0.25 < x < 9 to prevent damage to the components. Solving this inequality helps you determine the safe operating range for the voltage. In this case, the voltage must be less than -0.25 volts, which is not physically realizable in most circuits, indicating a design flaw or an impossible condition to satisfy within the given constraints. This type of problem helps in setting boundaries and understanding limitations in practical applications.
The electric device with a current of 15.0 A for 30 seconds delivers approximately 450 coulombs of charge. This charge corresponds to about 2.81 x 10^21 electrons flowing through it. Hence, around 2.81 x 10^21 electrons pass through the device in that time period.
;
