Problem 48 - Prealgebra

Question

Solve each inequality. Check your solution. $$\frac{n}{-4} \leq-11$$

Step-by-Step Solution

Answer

The solution is \( n \geq 44 \).

Step 1: Understand the Inequality

We are given the inequality \( \frac{n}{-4} \leq -11 \). Our goal is to find all values of \( n \) that satisfy this inequality.

Step 2: Multiply Both Sides by -4

To isolate \( n \), we need to eliminate the division by \(-4\). Multiply both sides of the inequality by \(-4\), remembering that multiplying by a negative number reverses the inequality sign: \[ n \geq (-11) \times (-4) \] Simplify the right side: \[ n \geq 44 \]

Step 3: Interpret the Result

The solution to the inequality is all real numbers \( n \) that are greater than or equal to 44. So, \( n \geq 44 \).

Step 4: Check the Solution

To ensure our solution is correct, pick a value \( n = 44 \), to verify: \[ \frac{44}{-4} = -11 \] This satisfies the inequality \( \frac{n}{-4} \leq -11 \). Try a value of \( n > 44 \), such as 45, to ensure the inequality holds: \[ \frac{45}{-4} = -11.25 \] which is indeed less than \(-11\). Both cases confirm our solution is correct.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that compare two values using signs such as \(<, \leq, >, \geq\). Solving them involves finding the set of all possible values that satisfy the condition given by the inequality.

In the exercise, the inequality given is \(\frac{n}{-4} \leq -11\). The goal is to find all values of \(n\) that make this statement true. Solving starts through a series of transformative steps that simplify and isolate the variable.

The primary method involves using mathematical operations to manipulate the inequality until it is in a simple form where \(n\) is isolated on one side. Bear in mind that when solving inequalities, each step should maintain the balance or the fairness of the inequality just like in equations.

Multiplying by Negative Numbers

A unique aspect of solving inequalities is the effect of multiplication or division by negative numbers. The inequality \( \frac{n}{-4} \leq -11 \) involves division by a negative, which we must address by multiplying both sides by \(-4\).

Here's the important part: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This means \(\leq\) becomes \(\geq\) and vice versa.

For example, when we multiply both sides of \( \frac{n}{-4} \leq -11 \) by \(-4\), the sign switches from \(\leq\) to \(\geq\), resulting in \( n \geq 44 \). This reversal is crucial and can be a common point of confusion, so always remember to switch the sign in these scenarios to maintain the correct relationship.

Interpreting Inequalities

After solving an inequality, understanding the solution is essential. The result \( n \geq 44 \) tells us that the solution includes all real numbers \(n\) that are greater than or equal to \(44\).

When interpreting inequalities:

• The inequality symbol: \(\geq\) means that 44 is included in the solution, unlike \(>\) where it would not be.
• Visualizing solutions: The solution set can be visualized on a number line, where numbers to the right of 44 (including 44 itself) are valid solutions.
• Verification: Always verify by substituting a few values back into the original inequality to ensure they satisfy it.

By understanding this interpretation, we can apply such concepts to various real-life scenarios where inequalities come into play, such as financial planning or statistical analysis.