2. $x$ times 3 minus 4 is greater than $x$ and less than 8. Find the range of $x$.
Answer (2)
Translate the problem into a compound inequality: x < 3 x − 4 < 8 .
Solve the first inequality x < 3 x − 4 , which simplifies to 2"> x > 2 .
Solve the second inequality 3 x − 4 < 8 , which simplifies to x < 4 .
Combine the results to express the range of x : 2 < x < 4 .
Explanation
Understanding the Problem Let's break down the problem. We are given that ' x times 3 minus 4 is greater than x and less than 8'. We need to find the range of possible values for x . This means we need to find both a lower limit and an upper limit for x .
Translating to Inequality First, we translate the given statement into a compound inequality. The statement ' x times 3 minus 4 is greater than x and less than 8' can be written as: x < 3 x − 4 < 8
Splitting the Inequality Now, we solve the compound inequality x < 3 x − 4 < 8 . This inequality can be split into two separate inequalities: x < 3 x − 4 and 3 x − 4 < 8
Solving the First Inequality Let's solve the first inequality, x < 3 x − 4 . To isolate x , we first subtract x from both sides: 0 < 2 x − 4 Then, we add 4 to both sides: 4 < 2 x Finally, we divide both sides by 2: 2 < x So, 2"> x > 2 .
Solving the Second Inequality Next, let's solve the second inequality, 3 x − 4 < 8 . To isolate x , we first add 4 to both sides: 3 x < 12 Then, we divide both sides by 3: x < 4
Combining the Inequalities Now, we combine the two inequalities to find the range of x . We have 2"> x > 2 and x < 4 . Therefore, the range of x is 2 < x < 4 .
Final Answer So, the range of x is between 2 and 4.
Examples
Understanding ranges like this is crucial in many real-world scenarios. For example, if you're designing a bridge, you need to ensure the materials can withstand a certain range of temperatures or weights. Similarly, in economics, understanding the range of possible outcomes for an investment helps in making informed decisions. This problem demonstrates how to define and solve for a range, which is a fundamental skill in both engineering and finance.
The range of values for x that satisfies the inequality is between 2 and 4, expressed as 2 < x < 4 .
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