Problem 32 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$-\frac{1}{3} x \geq-9$$

Step-by-Step Solution

Answer

The solution is \(x \leq 27\) and is verified by substitution.

Step 1: Isolate the Variable

To solve the inequality, start by isolating the variable on one side. Since you have \[-\frac{1}{3}x \geq -9\], you need to get rid of the fraction by multiplying both sides of the inequality by -3. Remember, multiplying or dividing an inequality by a negative number reverses the inequality sign. This gives \[x \leq 27\].

Step 2: Verify the Solution

Check your solution by choosing a value for \(x\) that satisfies \(x \leq 27\). For instance, if \(x = 27\), substitute back into the original equation: \[-\frac{1}{3}(27) = -9\], confirming \[-9 \geq -9\]. The solution holds true.

Step 3: Graph the Solution

To graph \(x \leq 27\) on a number line, draw a line and a closed circle (because 27 is included in the solution) at 27, then shade everything to the left to represent that \(x\) can take any value less than or equal to 27.

Key Concepts

Isolate the Variable

Solving inequalities often begins with isolating the variable. To isolate the variable means to rearrange the inequality so that the variable gets to stand alone on one side of the inequality sign. Here, in the inequality \[-\frac{1}{3}x \geq -9\], our target is to get \(x\) by itself. We achieve this by eliminating the fraction.

• First, multiply both sides of the inequality by -3, because -3 is the reciprocal of \(-\frac{1}{3}\). This helps to eliminate the fraction.
• Next, remember that when we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign. Therefore, the inequality changes from \(\geq\) to \(\leq\).

After these steps, we arrive at the simplified inequality: \[x \leq 27\]This tells us that \(x\) can be any number less than or equal to 27.

Inequality Reversal

A key concept in solving inequalities is understanding how and when the inequality sign reverses. The inequality sign will flip any time you multiply or divide both sides of an inequality by a negative number. This is a unique characteristic that distinguishes inequalities from equations. Let's consider our example further:

• When we multiplied both sides of \[-\frac{1}{3}x \geq -9\] by -3 to isolate \(x\), we needed to reverse the inequality sign because -3 is a negative number.
• Thus, the original \(\geq\) inequality sign turns into \(\leq\) in the resulting inequality \(x \leq 27\).

This flipping of the sign is crucial; neglecting it would result in a completely different and incorrect solution.

Number Line Graphing

Once you have your solution for an inequality, representing it visually on a number line helps solidify your understanding. For the inequality \(x \leq 27\), here's how you can show it on a number line:

• First, draw a horizontal line and mark a point at 27 on this line. Since 27 is included in the solution (\(x\) can be equal to 27), you should use a closed circle at 27.
• Next, shade all the area to the left of 27 to indicate that any value less than 27 is also a part of the solution set. This shaded region clearly demonstrates that \(x\) can be any number up to and including 27.

Using a number line is a wonderful visualization that can make understanding the range and flexibility of solutions in an inequality more intuitive.