What value of $x$ is in the solution set of $2x-3>11-5x$?
Answer (1)
Solve the inequality 11 - 5x"> 2 x − 3 > 11 − 5 x for x .
Add 5 x to both sides and then add 3 to both sides to get 14"> 7 x > 14 .
Divide both sides by 7 to find 2"> x > 2 .
From the given options, only x = 4 satisfies 2"> x > 2 , so the answer is 4 .
Explanation
Understanding the problem We are given the inequality 11 - 5x"> 2 x − 3 > 11 − 5 x and the possible values for x are -3, 0, 2, and 4. We need to find which of these values satisfy the inequality.
Isolating x term First, let's solve the inequality for x . Add 5 x to both sides: 11 - 5x + 5x"> 2 x − 3 + 5 x > 11 − 5 x + 5 x 11"> 7 x − 3 > 11
Further isolating x term Now, add 3 to both sides: 11 + 3"> 7 x − 3 + 3 > 11 + 3 14"> 7 x > 14
Solving for x Divide both sides by 7: \frac{14}{7}"> 7 7 x > 7 14 2"> x > 2
Checking the options Now we check the given values for x :
If x = − 3 , then 2"> − 3 > 2 is false. If x = 0 , then 2"> 0 > 2 is false. If x = 2 , then 2"> 2 > 2 is false. If x = 4 , then 2"> 4 > 2 is true.
Final Answer Therefore, the only value of x from the given options that satisfies the inequality is x = 4 .
Examples
Imagine you're determining the minimum sales target you need to exceed to earn a bonus. If your bonus condition is represented by the inequality 11 - 5x"> 2 x − 3 > 11 − 5 x , where x is the number of sales, solving this inequality helps you find the minimum number of sales ( 2"> x > 2 ) required to qualify for the bonus. This type of problem is useful in setting goals and understanding thresholds in various real-life scenarios, such as sales targets, performance metrics, or resource allocation.
