Calcula los elementos del siguiente conjunto
[tex]$B={2 x+1 / 4\ \textless \ x\ \textless \ 9}$[/tex]
Answer (1)
Solve the inequality 2 x + 1/4 < x , which gives x < − 1/4 .
State the inequality x < 9 .
Combine the two inequalities to find the solution set x < − 1/4 .
Express the solution set in interval notation: ( − ∞ , − 1/4 ) . The elements of the set B are all x such that x < − 1/4 .
Explanation
Understanding the Problem We are given the set B = { 2 x + 1/4 < x < 9 } . We need to find the values of x that satisfy the compound inequality 2 x + 1/4 < x < 9 . This means we need to solve two inequalities: 2 x + 1/4 < x and x < 9 .
Solving the First Inequality First, let's solve the inequality 2 x + 1/4 < x . Subtracting 2 x from both sides gives 1/4 < x − 2 x , which simplifies to 1/4 < − x . Multiplying both sides by − 1 and flipping the inequality sign gives x"> − 1/4 > x , or x < − 1/4 .
Stating the Second Inequality Next, we have the inequality x < 9 .
Combining the Inequalities Now, we need to find the values of x that satisfy both x < − 1/4 and x < 9 . Since − 1/4 is smaller than 9 , the intersection of these two inequalities is x < − 1/4 . Therefore, the solution set is all x such that x < − 1/4 .
Expressing the Solution Set The set B consists of all x such that x < − 1/4 . In interval notation, this is the interval ( − ∞ , − 1/4 ) .
Examples
Consider a scenario where you are designing a simple electrical circuit. The voltage V in the circuit must satisfy the condition 2 V + 0.25 < V < 9 for the circuit to function correctly. Solving this inequality helps determine the range of voltages that will allow the circuit to operate as intended. This type of problem is also applicable in various fields such as physics, engineering, and economics, where constraints are often expressed as inequalities.
