Verified Answer :

  • Isolate the absolute value term by adding \(9\) to both sides and dividing by \(3\), resulting in \(|m+7| < 13\).
  • Apply the property of absolute value inequalities to set up the compound inequality: \(-13 < m+7 < 13\).
  • Solve the compound inequality by subtracting \(7\) from all parts to isolate \(m\).
  • Conclude that the solution set is \(\boxed{-20 < m < 6}\).

Explanation

  1. Analyzing the inequality

The given inequality is \(3|m+7|-9 < 30\). We are dealing with an absolute value inequality involving a single variable \(m\). Our goal is to isolate \(m\) to determine the interval of values that satisfy the condition. The expression \(|m+7|\) represents the distance between \(m\) and \(-7\) on the real number line, and the properties of absolute value will allow us to break this down into a simpler compound inequality.

  1. Isolating the absolute value expression

First, we isolate the absolute value term. We begin by adding \(9\) to both sides of the inequality:
\[3|m+7| < 30 + 9\]
\[3|m+7| < 39\]
Next, we divide both sides by \(3\):
\[|m+7| < \frac{39}{3}\]
\[|m+7| < 13\]

  1. Removing the absolute value signs

Now that the absolute value is isolated, we use the property that for any \(a > 0\), the inequality \(|X| < a\) is equivalent to the compound inequality \(-a < X < a\). Applying this to our expression with \(X = m+7\) and \(a = 13\), we get:
\[-13 < m+7 < 13\]

  1. Solving for the variable \(m\)}],example_of_usage:

Finally, we solve for \(m\) by subtracting \(7\) from all parts of the compound inequality:
\[-13 - 7 < m + 7 - 7 < 13 - 7\]
\[-20 < m < 6\]
This result defines the range of values for \(m\) that satisfy the original inequality. Thus, the correct solution is \(-20 < m < 6\).

  1. Real-life example of absolute value inequalities

Imagine a factory produces precision metal rods that must have a length of exactly \(7\) cm to fit into a machine, but there is a allowed tolerance in the manufacturing process. The acceptable range for the deviation \(m\) from the target can be modeled by the inequality \(3|m+7|-9 < 30\). By solving this, we find that the deviation \(m\) must fall strictly between \(-20\) and \(6\). Understanding this range is crucial for quality control in manufacturing, ensuring that all components fit perfectly and the machinery operates safely and efficiently.

  1. Summary of the solution

The solution is:

  • Isolate the absolute value term by adding \(9\) to both sides and dividing by \(3\), resulting in \(|m+7| < 13\).
  • Apply the property of absolute value inequalities to set up the compound inequality: \(-13 < m+7 < 13\).
  • Solve the compound inequality by subtracting \(7\) from all parts to isolate \(m\).
  • Conclude that the solution set is \(\boxed{-20 < m < 6}\).

Examples

Imagine a factory produces precision metal rods that must have a length of exactly \(7\) cm to fit into a machine, but there is a allowed tolerance in the manufacturing process. The acceptable range for the deviation \(m\) from the target can be modeled by the inequality \(3|m+7|-9 < 30\). By solving this, we find that the deviation \(m\) must fall strictly between \(-20\) and \(6\). Understanding this range is crucial for quality control in manufacturing, ensuring that all components fit perfectly and the machinery operates safely and efficiently.

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