Problem 35 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$\frac{k}{-2}<9$$

Step-by-Step Solution

Answer

Solution: \( k > -18 \). Graph: Open circle at \(-18\), shade rightwards.

Step 1: Multiply Both Sides by -2

The inequality given is \( \frac{k}{-2} < 9 \). To eliminate the fraction, multiply both sides of the inequality by \(-2\). Remember, when multiplying or dividing an inequality by a negative number, the inequality sign reverses. So, we have: \[ k > 9 imes (-2) \]},

Step 2: Calculate the Right Side

Now, calculate the right side of the inequality: \[ 9 imes (-2) = -18 \]. Substitute this back into the inequality from Step 1: \[ k > -18 \]

Step 3: Verify the Solution

To verify, pick a number greater than \(-18\), say \(-17\), and substitute back into the original inequality.\( \frac{-17}{-2} = 8.5 \), which is indeed less than 9. This confirms that our solution \( k > -18 \) is correct.

Step 4: Graph the Solution

Draw a number line and place an open circle at \(-18\) to indicate that \(-18\) is not included in the solution. Shade the entire region to the right of \(-18\) to represent all values greater than \(-18\).

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers in a straight line. It can be very helpful when solving inequalities, as it allows you to easily see where numbers fall in relation to each other.
Using a number line, you can graphically show the solution to an inequality by marking the key number and indicating which side of it contains the solutions.
In the inequality \( k > -18 \), you place an open circle at \(-18\) because \(-18\) is not part of the solution (as it only counts numbers greater than \(-18\)).
Then, you shade everything to the right of \(-18\) to show the solution set. This shaded area represents all possible values of \( k \) that satisfy the inequality.

Checking Solutions in Inequalities

Checking the solution of an inequality is crucial to ensure the correctness of your answer. This is done by substituting a number from the solution set back into the original inequality to see if it holds true.
In our example \( \frac{k}{-2} < 9 \), once we solved it to find \( k > -18 \), a good check would be to choose a number like \(-17\).
Substituting \(-17\) into the original inequality gives \( \frac{-17}{-2} = 8.5 \), which is indeed less than 9.
This confirms that our solution is correct. If the chosen number does not satisfy the inequality, you should revisit your calculations and steps to find the mistake.
Always make sure to check your solutions to avoid small errors.

Multiplying or Dividing Inequalities

When dealing with inequalities, a critical rule is how the inequality sign changes while multiplying or dividing by a negative number. If you multiply or divide both sides of an inequality by a negative, the direction of the inequality sign reverses.
For example, given \( \frac{k}{-2} < 9 \), we need to eliminate the fraction by multiplying both sides by \(-2\).
Doing so reverses the inequality sign, turning it into \( k > -18 \).
It’s easy to forget to flip the sign, so always remember this step. It’s a simple but crucial part of handling inequalities correctly.

Graphing Inequalities

Graphing inequalities on a number line is a simple yet effective way to visualize solutions.
Once you’ve solved the inequality, like \( k > -18 \) in our example, you start by marking \(-18\) on the number line.
Use an open circle because \(-18\) itself is not part of the solution.
Then, shade the right side of \(-18\) to show all numbers greater than \(-18\).

• An open circle means the number itself is not included (\( > \) or \( < \)).
• A closed circle includes the number (\( \geq \) or \( \leq \)).

Shade in the correct direction based on the inequality sign, ensuring accuracy.