Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. $$0 \leq \frac{x+3}{2}<5$$

Step-by-Step Solution

Answer

The solution to the inequality \(0 \leq\frac{x+3}{2}<5\) is \(-3 \leq x < 7\)

Step 1: Simplify the Inequality

Begin by simplifying the inequality. The main aim in this step is to isolate \(x\) which is in the numerator. One way of doing this is by multiplying the entire inequality by the denominator which is \(2\). This gives: \(0*2 \leq (x+3) < 5*2\) Hence the simplified inequality becomes: \(0 \leq x+3 <10\)

Step 2: Solve for x

Next, solve for \(x\). This can be achieved by subtracting \(3\) from all parts of the inequality; this leads to: \(0-3 \leq x+3-3 < 10-3\) which simplifies to: \(-3 \leq x < 7\)

Step 3: Graphical Representation

Now that the solution for \(x\) has been obtained, it can be represented on a number line graph. The range of \(x\) extends from \(-3\) to \(7\), including \(-3\) but excluding \(7\). The included beginning point \(-3\) should be represented by a filled circle on the number line, whereas the excluded end point \(7\) is represented by an open circle. The section between these two points is a continuous line that represents the solution set.

Step 4: Use a Graphing Utility for Verification

Finally, use a graphing utility to check the solution. The graphical representation should match the theoretical solution obtained. By plugging the inequality into the software, the result should display a graph similar to the one sketched in the previous step.

Key Concepts

Number Line Graphing

Number line graphing is a helpful visual method to display solutions to inequalities. In this context, the solution to the inequality \(-3 \leq x < 7\) must be represented visually. A number line is a straight line with numerical values placed sequentially from left to right.

• Begin by drawing a horizontal line, marking points for at least the beginning (-3) and ending (7) values.
• Mark \-3\ with a filled circle to show it is included in the solution set. The filled circle represents that \(-3\) is part of the solution.


• Mark 7 with an open circle to indicate it is not part of this solution. The open circle shows exclusion.
• Finally, draw a solid line, starting from the filled circle at \(-3\), extending towards but not touching the open circle at 7.

This visual representation through a number line helps highlight which values meet the criteria set by the inequality, providing a clear and intuitive comprehension of the solution set.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that make the inequality true. The exercise presents the inequality \(0 \leq \frac{x+3}{2} < 5\). To solve it, you need to isolate \(x\) on one side.

• First, multiply each part of the inequality by 2 to eliminate the fraction. This gives \(0 \leq x + 3 < 10\).


• Next, subtract 3 from each part to further isolate \(x\), simplifying the expression to \(-3 \leq x < 7\).

Each step in solving an inequality is crucial for maintaining the integrity of the solution. Remember, when multiplying or dividing by negative values in inequalities, the inequality signs must be flipped. Thankfully, in this problem, all operations involve positive numbers, so no flipping is necessary. Mastery of these steps ensures a precise solution to any similar inequality challenge.

Graphing Utility Verification

A graphing utility can be used to verify the solution derived from solving an inequality. This tool helps visualize the solution set and confirm accuracy visually. Here’s how to use it effectively:

• Enter the inequality \(-3 \leq x < 7\) into the graphing utility.


• Adjust graph settings to appropriately display the number line or interval representation.


• The graph should illustrate the region between \(-3\) and 7; you'll see a filled mark at \(-3\) and an open mark at 7.

If the graph coincides with your number line sketch, it confirms the solution is correct. This graphic verification serves as a second check to ensure the problem-solving process was done accurately. Harnessing technology in this way enhances understanding and builds confidence in solving inequalities.