Question

Solve the inequality. Then graph the solution set on the real number line. \(\frac{3}{2} x \geq 9\)

Short answer

The solution to the inequality is \(x \in (0, 5)\). This can be represented graphically on the number line with a closed circle at x = 0 and an open circle at x = 5, with the line shaded in between.

Step-by-step solution

- Analysis

First analyze the inequality. The denominator should not equal zero, that means \(x \neq 5\). Keep this in mind for the final solution.

- Simplify Inequality

Bring the inequality to a simpler form, for this subtract 4 from both sides of the inequality. So, it becomes \(\frac{3x -5 - 4(x -5)}{x - 5} > 0\), which simplifies to \(\frac{-x}{x - 5} > 0\).

- Determine Critical Points

Critical points can be found by converting the inequality into an equation and solving it. That provides the values that make the numerator equal to zero and the values that make the denominator equal to zero. So we have: -x = 0 which gives x = 0 and x - 5 = 0, which gives x = 5.

- Test Intervals

We have three intervals to test based on the critical points: (-∞, 0), (0, 5), (5, ∞). Choose a test point in each interval and substitute it into the simplified inequality from Step 2. For (-∞, 0) choose -1, for (0, 5) choose 2, for (5, ∞) choose 6. After substituting, we find: \(\frac{-(-1)}{-1 - 5} = \frac{1}{-4} < 0\), \(\frac{-(2)}{2 - 5} = \frac{-2}{-3} > 0\), \(\frac{-(6)}{6 - 5} = -6 < 0\). So the inequality is satisfied in the intervals (0, 5).

- Graph the Solution

Plot this solution on a number line. Remember that x ≠ 5, so that point must be left open on the number line. This interval represents the solution set for the given inequality.

Key concepts

The Real Number Line

The real number line is a visual representation of all possible numbers from negative infinity to positive infinity. It helps in understanding where certain numbers or solutions are located.
On the number line, each point represents a distinct real number, such as -2, 0, or 3.
When graphing solutions to inequalities, the real number line becomes essential. It shows which parts of the line satisfy the inequality.
Typically, a filled circle is used for numbers included in a solution, while an open circle represents numbers that are not included.
For example, solutions where the inequality is strict (>) are shown with open circles to indicate that those points are not part of the solution itself.

Understanding Critical Points

Critical points in inequalities are the numbers that might make the inequality switch from true to false or vice versa.
These are crucial to identify because they help in dividing the number line into different sections, known as intervals.
In our exercise, we identified critical points by converting the inequality into an equation. We found two critical points: 0 and 5.
One critical point, where the numerator equals zero, helps show where the expression could potentially change sign.
The other, found by setting the denominator to zero, is essential as it shows where the expression might be undefined.
Remember that such critical points are used to establish boundaries on the number line for further testing.

Evaluating Test Intervals

Once critical points are determined, the real number line is divided into test intervals.
Each interval needs a test point to check whether the inequality holds.
In the exercise, this process broke the number line into three intervals: (-∞, 0), (0, 5), and (5, ∞).
For each interval, we selected a convenient test point like -1, 2, and 6, and substituted it back into the inequality.
The result determined which intervals satisfied the inequality, in this case, (0, 5).
By testing these intervals, we ensure that we cover all possible solutions.

Graphing Solutions on the Number Line

Graphing solutions is the final step that visually represents the solution on the number line.
In our exercise, after determining the valid interval, the next step is to draw it.
We start by marking 0 with an open circle as it satisfies the inequality but should not include the endpoint.
Then, draw a line connecting it and stop just before 5, leaving it open since it's undefined at x = 5.
This graphical representation makes interpreting the solution straightforward and understandable.

• Open circle at 0 shows the starting point.
• Line from 0 to just before 5 highlights the solution interval.
• Open circle at 5 correctly shows exclusion of that point.

This method helps in quickly identifying valid solutions to any given inequality.