Solve the equation. If there is no solution, select no solution.
[tex]5|n|-15=40[/tex]
Answer (2)
Isolate the absolute value term, split into cases, and solve for n. The solutions are n = 11 and n = − 11 . Therefore, the final answer is n = 11 or n = − 11 .
Explanation
Understanding the Problem We are given the equation 5∣ n ∣ − 15 = 40 and we need to solve for n . This equation involves an absolute value, so we will need to consider two cases.
Isolating the Absolute Value First, we isolate the absolute value term by adding 15 to both sides of the equation: 5∣ n ∣ − 15 + 15 = 40 + 15
5∣ n ∣ = 55
Solving for |n| Next, we divide both sides of the equation by 5 to isolate the absolute value: 5 5∣ n ∣ = 5 55
∣ n ∣ = 11
Considering Both Cases Now we consider two cases:
Case 1: n ≥ 0 . In this case, ∣ n ∣ = n , so we have n = 11 . Since 11 ≥ 0 , this is a valid solution.
Case 2: n < 0 . In this case, ∣ n ∣ = − n , so we have − n = 11 . Multiplying both sides by -1, we get n = − 11 . Since − 11 < 0 , this is a valid solution.
Finding the Solutions Therefore, the solutions are n = 11 and n = − 11 .
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering. For example, if you are manufacturing a part that needs to be exactly 5 cm long, but you allow for a tolerance of 0.1 cm, you can express this as an absolute value equation: ∣ x − 5∣ ≤ 0.1 , where x is the actual length of the part. Solving this inequality will give you the range of acceptable lengths for the part.
The solutions to the equation 5∣ n ∣ − 15 = 40 are n = 11 and n = − 11 . This involves isolating the absolute value and considering both positive and negative cases. Thus, both solutions are valid for the given equation.
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