Answered

Solve the inequality below, then choose the correct graph that represents the answer.

$2 n+7 \geq 27 \text { or } 3+3 n \leq 30$

Verified Answer :

  • Solve the first inequality \(2n + 7 \geq 27\) to get \(n \geq 10\).
  • Solve the second inequality \(3 + 3n \leq 30\) to get \(n \leq 9\).
  • Combine the results using the "or" operator, meaning the union of the two sets.
  • The final solution is \(\boxed{n \leq 9 \text{ or } n \geq 10}\).

Explanation

  1. Understanding the Problem

The problem asks us to solve the compound inequality given by:
\[2n + 7 \geq 27 \quad \text{or} \quad 3 + 3n \leq 30.\]
The use of the logical "or" implies that our solution will be the union of the sets of values that satisfy either of the two inequalities individually. We will solve each inequality separately and then combine their solution sets.

  1. Solving the First Inequality

First, we solve the inequality \(2n + 7 \geq 27\):

  1. Subtract 7 from both sides: \(2n \geq 27 - 7\), which simplifies to \(2n \geq 20\).
  2. Divide both sides by 2: \(n \geq \frac{20}{2}\), which results in \(n \geq 10\).

This inequality is satisfied for all values of \(n\) that are greater than or equal to 10.

  1. Solving the Second Inequality

Next, we solve the inequality \(3 + 3n \leq 30\):

  1. Subtract 3 from both sides: \(3n \leq 30 - 3\), which simplifies to \(3n \leq 27\).
  2. Divide both sides by 3: \(n \leq \frac{27}{3}\), which results in \(n \leq 9\).

This inequality is satisfied for all values of \(n\) that are less than or equal to 9.

  1. Combining the Results

The solution to the entire problem is the set of all values \(n\) such that \(n \geq 10\) or \(n \leq 9\).
In interval notation, this is \((-\infty, 9] \cup [10, \infty)\).
Graphically, this represents all numbers on the number line except for those between 9 and 10 (exclusive). This consists of two rays: one pointing to the left from 9 (inclusive), and one pointing to the right from 10 (inclusive). Since every real number is either \(\leq 9\), \(\geq 10\), or in the gap \((9, 10)\), the solution effectively covers all real numbers except the open interval \((9, 10)\).

Examples

Compound inequalities are common when describing safe operating ranges for machinery. For example, if a machine's temperature \(T\) must be kept either below \(90^\circ\text{C}\) for quick processes or above \(100^\circ\text{C}\) for high-efficiency modes, but never between \(90^\circ\text{C}\) and \(100^\circ\text{C}\) to avoid condensation issues, the safety condition would be \(T \leq 90 \text{ or } T \geq 100\). This ensures the device operates safely by avoiding the problematic intermediate range.

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