Use the graph to determine the solution of the inequality [tex] |x+1|+2 \textgreater \ 5 [/tex].
Answer (2)
The solution to the inequality 5"> ∣ x + 1∣ + 2 > 5 is found to be ( − ∞ , − 4 ) ∪ ( 2 , ∞ ) . This means that the values of x can be less than -4 or greater than 2. Hence, the final answer is ( − ∞ , − 4 ) ∪ ( 2 , ∞ ) .
;
Isolate the absolute value term: 3"> ∣ x + 1∣ > 3 .
Split the absolute value inequality into two cases: 3"> x + 1 > 3 and x + 1 < − 3 .
Solve each case: 2"> x > 2 and x < − 4 .
Express the solution in interval notation: ( − ∞ , − 4 ) ∪ ( 2 , ∞ ) .
Explanation
Understanding the Problem We are given the inequality 5"> ∣ x + 1∣ + 2 > 5 . Our goal is to find the solution set for x . The provided options are:
( − ∞ , − 4 ) ∪ ( 2 , ∞ )
( − 4 , 2 )
( 0 , 3 )
( − ∞ , 0 ) ∪ ( 3 , ∞ )
Isolating the Absolute Value First, we need to isolate the absolute value term. We subtract 2 from both sides of the inequality: 5 - 2"> ∣ x + 1∣ > 5 − 2 3"> ∣ x + 1∣ > 3
Considering Two Cases Now, we consider two cases based on the properties of absolute values:
Case 1: 3"> x + 1 > 3 Case 2: x + 1 < − 3
Solving the Cases Let's solve each case:
Case 1: 3"> x + 1 > 3 . Subtract 1 from both sides: 3 - 1"> x > 3 − 1 2"> x > 2
Case 2: x + 1 < − 3 . Subtract 1 from both sides: x < − 3 − 1 x < − 4
Combining the Solutions Combining the solutions from both cases, we have 2"> x > 2 or x < − 4 . In interval notation, this is: ( − ∞ , − 4 ) ∪ ( 2 , ∞ )
Final Answer Comparing our solution with the given options, we find that it matches option 1:
( − ∞ , − 4 ) ∪ ( 2 , ∞ )
Examples
Absolute value inequalities are useful in various real-world scenarios. For example, consider a manufacturing process where a machine needs to cut a metal rod to a specific length, say 10 cm. Due to the machine's limitations, the actual length might deviate by a certain tolerance, say 0.5 cm. This situation can be modeled using an absolute value inequality: ∣ x − 10∣ < 0.5 , where x is the actual length of the rod. Solving this inequality helps determine the acceptable range of lengths for the metal rods, ensuring they meet the required specifications. Similarly, in finance, absolute value inequalities can be used to model acceptable deviations in investment returns or budget allocations.
