If $x$ lies between 1 and 2, but does not equal 1 or 2, which of these statements is true?
A. $1< x +1<2$
B. $1
C. $-1<2-x<1$
D. $-1<3-2 x<1$
Answer (1)
Analyze each option by isolating x in the inequality.
Option A simplifies to 0 < x < 1 , which is false.
Option B simplifies to 2 < x < 3 , which is false.
Option C simplifies to 1 < x < 3 , which is true, but less precise.
Option D simplifies to 1 < x < 2 , which is exactly the given condition.
Therefore, the correct answer is D .
Explanation
Understanding the Problem We are given that x lies between 1 and 2, which means 1 < x < 2 . We need to determine which of the given inequalities is true. Let's analyze each option separately.
Analyzing Option A Option A: 1 < x + 1 < 2 . To isolate x , we subtract 1 from all parts of the inequality: 1 − 1 < x + 1 − 1 < 2 − 1 , which simplifies to 0 < x < 1 . This contradicts the given condition 1 < x < 2 , so option A is false.
Analyzing Option B Option B: 1 < x − 1 < 2 . To isolate x , we add 1 to all parts of the inequality: 1 + 1 < x − 1 + 1 < 2 + 1 , which simplifies to 2 < x < 3 . This contradicts the given condition 1 < x < 2 , so option B is false.
Analyzing Option C Option C: − 1 < 2 − x < 1 . To isolate x , we first subtract 2 from all parts of the inequality: − 1 − 2 < 2 − x − 2 < 1 − 2 , which simplifies to − 3 < − x < − 1 . Now, we multiply all parts of the inequality by -1, remembering to reverse the inequality signs: (-x) "> (-1)"> ( − 3 ) " > ( − x ) " > ( − 1 ) , which gives us 1 < x < 3 . The given condition is 1 < x < 2 . Since 1 < x < 2 is a subset of 1 < x < 3 , option C is true, but not as precise as option D.
Analyzing Option D Option D: − 1 < 3 − 2 x < 1 . To isolate x , we first subtract 3 from all parts of the inequality: − 1 − 3 < 3 − 2 x − 3 < 1 − 3 , which simplifies to − 4 < − 2 x < − 2 . Now, we divide all parts of the inequality by -2, remembering to reverse the inequality signs: \frac{-2x}{-2} > \frac{-2}{-2}"> − 2 − 4 > − 2 − 2 x > − 2 − 2 , which gives us x > 1"> 2 > x > 1 , or 1 < x < 2 . This matches the given condition exactly, so option D is true.
Conclusion Since option D gives us the exact condition 1 < x < 2 , it is the correct answer.
Examples
Understanding inequalities is crucial in various real-world scenarios. For instance, when designing a bridge, engineers must ensure that the materials used can withstand a certain range of stress and strain. If the stress exceeds the maximum limit or falls below the minimum limit, the bridge could be at risk of failure. Similarly, in economics, understanding inequalities helps in analyzing income distribution and poverty levels. By setting appropriate boundaries, policymakers can implement effective strategies to reduce income inequality and improve the living standards of the less privileged.
