What is the solution to the inequality $|2 n+5|>1$?
A. $-3>n>-2$
B. $2 -2$
D. $n<2$ or $n>3
Answer (2)
Split the absolute value inequality into two cases: 1"> 2 n + 5 > 1 and 2 n + 5 < − 1 .
Solve the first inequality: -4"> 2 n > − 4 , which gives -2"> n > − 2 .
Solve the second inequality: 2 n < − 6 , which gives n < − 3 .
Combine the solutions: n < − 3 or -2"> n > − 2 . The final answer is -2}"> n < − 3 or n > − 2 .
Explanation
Understanding the Problem We are given the inequality 1"> ∣2 n + 5∣ > 1 and asked to find the solution. The absolute value inequality can be split into two separate inequalities.
Objective We need to solve the inequality 1"> ∣2 n + 5∣ > 1 for n .
Solution Plan We will split the absolute value inequality into two separate inequalities: 1"> 2 n + 5 > 1 and 2 n + 5 < − 1 .
Solving the First Inequality Let's solve the first inequality 1"> 2 n + 5 > 1 for n . Subtract 5 from both sides: 1-5"> 2 n > 1 − 5 , which simplifies to -4"> 2 n > − 4 . Divide both sides by 2: -2"> n > − 2 .
Solving the Second Inequality Now, let's solve the second inequality 2 n + 5 < − 1 for n . Subtract 5 from both sides: 2 n < − 1 − 5 , which simplifies to 2 n < − 6 . Divide both sides by 2: n < − 3 .
Combining the Solutions Finally, we combine the solutions to the two inequalities. The solution is n < − 3 or -2"> n > − 2 .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, in manufacturing, the dimensions of a product must be within a certain tolerance of the specified measurement. If the specified length of a metal rod is 10 cm, and the tolerance is 0.1 cm, then the actual length x must satisfy the inequality ∣ x − 10∣ < 0.1 . This means the length can be between 9.9 cm and 10.1 cm. Similarly, in finance, one might use absolute value inequalities to describe the acceptable range of investment returns or to control risk.
To solve the inequality 1"> ∣2 n + 5∣ > 1 , we split it into two cases which lead us to find that n < − 3 or -2"> n > − 2 . The correct answer is option C. This involves using properties of absolute values and basic algebra to find the solution set.
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