Problem 61 - Algebra 2

Question

Solve each inequality. (lessons \(1-5\) and \(1-6 )\) $$ |y+1| < 7 $$

Step-by-Step Solution

Answer

The solution is \(-8 < y < 6\).

Step 1: 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: \(-7 < y+1 < 7\).

Step 2: 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 \(-8 < y\).

Step 3: 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 \(y < 6\).

Step 4: Combine the Inequalities

Combine the solutions from Steps 2 and 3 to express the solution to the original compound inequality. The solution is \(-8 < y < 6\).

Step 5: 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.

Key Concepts

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:

• For \(|A| < B\), it translates to \(-B < A < B\).
• For \(|A| > B\), it simplifies to either \(A > B\) or \(A < -B\).

In this exercise, transforming \(|y+1| < 7\) becomes a compound inequality: \(-7 < y+1 < 7\). By solving these inequalities separately, we find the range of values y can take to satisfy the condition.

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.

• 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 \(y < 6\).

The solution to the compound inequality is the overlap of these two solutions, or the range \(-8 < y < 6\).

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).

• Draw a line connecting the section between \(-8\) and \(6\).

This segment clearly represents all the possible values of \(y\) that satisfy both parts of the compound inequality, visually showing all potential solutions.

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:

• 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.