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Magazine Chapter 2: Problem 61
Solve each inequality. (lessons \(1-5\) and \(1-6 )\) $$ |y+1| < 7 $$
Short Answer
Expert verified
The solution is \(-8 < y < 6\).
Step by step solution
01
Understanding Absolute Value Inequality
The inequality \(|y+1| < 7\) means that the expression inside the absolute value, \(y+1\), is within 7 units of 0 on the number line. This translates to two separate inequalities: \(-7 < y+1 < 7\).
02
Solving the First Inequality
Let's solve the first part of the inequality: \(-7 < y+1\). To isolate \(y\), subtract 1 from both sides, yielding \(-7 - 1 < y\). Simplifying gives us \(-8 < y\).
03
Solving the Second Inequality
Now solve the second part of the compound inequality: \(y+1 < 7\). Subtract 1 from both sides here as well, resulting in \(y < 7 - 1\). Simplifying, we have \(y < 6\).
04
Combine the Inequalities
Combine the solutions from Steps 2 and 3 to express the solution to the original compound inequality. The solution is \(-8 < y < 6\).
05
Verify the Solution
Choose a value within the solution interval, like \(y = 0\), and substitute it back into the original inequality: \(|0+1| = 1\). Since 1 is less than 7, the solution verifies correctly. You can check additional points to ensure the solution is accurate.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inequality Solving
To solve absolute value inequalities like \(|y+1| < 7\), we need to understand the nature of absolute values. An absolute value equation like this one indicates the distance of the expression inside from zero on a number line. For inequalities, this converts into two separate inequalities.
The general rule of thumb is:
The general rule of thumb is:
- For \(|A| < B\), it translates to \(-B < A < B\).
- For \(|A| > B\), it simplifies to either \(A > B\) or \(A < -B\).
Compound Inequalities
Compound inequalities involve solving more than one inequality at a time, often linked by 'and' or 'or'. When dealing with 'and', as in this exercise, you look for a common solution that satisfies both inequalities.
Our original inequality \(|y+1| < 7\) translates into two simpler inequalities \(-7 < y+1\) and \(y+1 < 7\). The task is to solve each inequality independently.
Our original inequality \(|y+1| < 7\) translates into two simpler inequalities \(-7 < y+1\) and \(y+1 < 7\). The task is to solve each inequality independently.
- For \(-7 < y+1\), subtract 1 from each side to isolate \(y\), leaving you with \(-8 < y\).
- For \(y+1 < 7\), the same process of subtracting 1 results in \(y < 6\).
Number Line Representation
A number line is a helpful tool to visualize compound inequalities. It allows you to see the range of solutions clearly and ensures you capture all valid solutions.
For \(-8 < y < 6\), mark open circles at \(y = -8\) and \(y = 6\) on the number line because these values are not included (the inequality is strictly less than and greater than).
For \(-8 < y < 6\), mark open circles at \(y = -8\) and \(y = 6\) on the number line because these values are not included (the inequality is strictly less than and greater than).
- Draw a line connecting the section between \(-8\) and \(6\).
Solution Verification
To ensure our solution to an inequality is correct, it's vital to verify by substituting values back into the original inequality.
Choosing a value within the interval can confirm correctness. For example, using \(y = 0\) from the range \(-8 < y < 6\), plug it back:\[|0 + 1| = 1\]Since \(1 < 7\), it works.
Try other values from within and outside the range:
Choosing a value within the interval can confirm correctness. For example, using \(y = 0\) from the range \(-8 < y < 6\), plug it back:\[|0 + 1| = 1\]Since \(1 < 7\), it works.
Try other values from within and outside the range:
- Inside the range: Try \(y = -7\) and \(y = 5\), both should satisfy \(|y+1| < 7\).
- Outside the range: Try \(y = -9\) or \(y = 7\), both should not satisfy the condition, confirming they are not part of the solution.
