Problem 26 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$\frac{n}{-5} \geq-0.8$$

Step-by-Step Solution

Answer

The solution is \(n \leq 4\).

Step 1: Remove the Fraction

To eliminate the fraction, multiply both sides by (-5). Remember that the inequality sign will switch direction when multiplying or dividing by a negative number. So, \(-5 \cdot \frac{n}{-5} \leq -5 \cdot (-0.8)\).This simplifies to:\[n \leq 4\]

Step 2: Solution Validation

To ensure our solution is correct, pick a value of \(n\) less than or equal to 4 and substitute it back into the original inequality. For example, if \(n = 0\), \(\frac{0}{-5} = 0\), and \(0 \geq -0.8\), which is true, confirming our solution.

Step 3: Graphing the Solution

To graph \(n \leq 4\) on a number line, draw a number line and place a filled circle on 4. Shade all the regions to the left of 4, indicating all numbers less than or equal to 4 are solutions.

Key Concepts

Fraction Operations

When dealing with inequalities involving fractions, a common approach is to eliminate the fraction to make calculations simpler.
One effective method is by multiplying both sides of the inequality by the denominator.

• This removes the fraction from the variable, making it easier to solve.
• Be mindful: when multiplying or dividing by a negative number, the inequality sign must flip direction.

In our original example, the inequality is \( \frac{n}{-5} \geq -0.8 \). To remove the fraction, multiply every term by -5, resulting in \( n \leq 4 \). Remember, the sign changes due to the negative multiplication!

Number Line Graphing

Graphing inequalities on a number line provides a clear visualization of all possible solutions.
This helps in understanding which values satisfy the inequality.

• A filled circle is used on the number line to indicate values that are included in the solution.
• An open circle is used when a value is not part of the solution.
• Shade the appropriate side of the number line to show all potential solutions.

In our solution, because \( n \leq 4 \), place a filled circle at 4, and shade the area extending leftwards. This represents all the numbers less than or equal to 4.

Inequality Solving

Solving inequalities is much like solving equations, but with key differences due to the inequality signs.
It involves isolating the variable on one side of the inequality by performing legal arithmetic operations, including:

• Adding or subtracting numbers.
• Multiplying or dividing, while being cautious of sign changes.

In our exercise, we multiplied by -5 to eliminate the fraction and flipped the sign, resulting in \( n \leq 4 \). Properly solving and testing values can confirm accuracy.

Negative Values in Inequalities

Working with negative numbers in inequalities requires special attention, especially when performing arithmetic operations.
A crucial aspect is that multiplying or dividing by a negative number reverses the inequality sign.

• This is a unique behavior limited to inequalities.
• Always double-check when negatives are involved to ensure your solution remains true.

For example, starting with \( \frac{n}{-5} \geq -0.8 \), multiplication by -5 required flipping the inequality to \( n \leq 4 \). This flipped sign is essential to balance the inequality accurately.