Question
Solve the following inequality, and write interval notation for the solution set. Then, graph the solution set. \[ |x+6|<2 \]
Ask by Pope Tyler. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution to the inequality |x + 6| < 2 is all numbers between -8 and -4. In interval notation, this is (-8, -4). On the number line, draw an open circle at -8 and -4, and shade the line between them.
Solution
To solve |x + 6| < 2, start by recognizing that the inequality means the distance between x + 6 and 0 is less than 2. This can be rewritten as a compound inequality:
-2 < x + 6 < 2
Next, isolate x by subtracting 6 from all parts:
-2 - 6 < x + 6 - 6 < 2 - 6
-8 < x < -4
The solution in interval notation is:
(-8, -4)
Graphing the solution on a number line:
• Draw a horizontal line and mark the points at -8 and -4.
• Place open circles at -8 and -4 (since the inequality is strict and does not include these endpoints).
• Shade the region between -8 and -4 to indicate all numbers in that interval.
Thus, the final answer is:
Solution set: (-8, -4)
Graph: A number line with an open circle at -8, an open circle at -4, and the line segment between them shaded.
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Simplify this solution Mind Expander
To solve the inequality \( |x+6| < 2 \), we need to break it down into two separate inequalities: 1. \( x + 6 < 2 \) 2. \( x + 6 > -2 \) Now, let's solve these inequalities: 1. From \( x + 6 < 2 \): \[ x < 2 - 6 \\ x < -4 \] 2. From \( x + 6 > -2 \): \[ x > -2 - 6 \\ x > -8 \] Now, we can combine these results. The solution set is: \[ -8 < x < -4 \] In interval notation, this is written as: \[ (-8, -4) \] To graph this solution set, you would draw a number line, placing open circles at \(-8\) and \(-4\), and shade the region between them to indicate that all values between \(-8\) and \(-4\) are included in the solution.
