Problem 20 - 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. $$3 x+1 \geq 2+x$$

Step-by-Step Solution

Answer

The solution to the inequality \(3x + 1 \geq 2 + x\) is \(x \geq \frac{1}{2}\). When this range of \(x\) values is represented on the number line, it starts from \(x = \frac{1}{2}\) and extends to the right, towards positive infinity. A graphing utility would confirm these results.

Step 1: Simplify the Inequality

Firstly, collect the \(x\) terms on one side and constants on the other side by subtracting \(x\) from both sides and subtracting 1 from both sides. The inequality is transformed as follows:\(3x - x \geq 2 - 1\)Which simplifies to,\(2x \geq 1\).

Step 2: Solve for x

To solve for \(x\), divide through by 2. This results to:\(x \geq \frac{1}{2}\).

Step 3: Represent on the Number Line

Plot the point \(\frac{1}{2}\) on the number line and because the inequality is 'greater than or equal to', indicate this point as a filled dot or circle. Everything to the right of this point satisfies the inequality, so draw an arrow going towards the right from this point.

Step 4: Use a Graphing Utility to Verify

A graph of this inequality would indicate a straight line crossing the x-axis at \(x = \frac{1}{2}\) and shading to the right. You can use a graphing utility to input the inequality and check that it matches this.

Key Concepts

Solving Linear Inequalities

When tackling linear inequalities such as \(3x + 1 \geq 2 + x\), the goal is to find the set of values for \(x\) that satisfy the inequality. Start by simplifying it to make the terms more manageable. Collect all terms involving \(x\) on one side, and move constants to the opposite side. In this exercise, subtract \(x\) and 1 from both sides to get \(2x \geq 1\).

Once simplified, the next step is to solve for \(x\) by isolating it. This entails dividing both sides by the coefficient of \(x\), which in this case is 2. Doing so gives \(x \geq \frac{1}{2}\).

Remember, dividing or multiplying both sides of an inequality by a positive number preserves the inequality sign. The solution \(x \geq \frac{1}{2}\) tells us that any number equal to or greater than 0.5 will satisfy the original inequality.

Graphing Inequality Solutions

After solving a linear inequality, graphing its solution helps visually confirm and understand the result. The equation \(x \geq \frac{1}{2}\) represents a straight line on a Cartesian plane, but we are more interested in depicting the range of solutions.

For the inequality \(x \geq \frac{1}{2}\), the graph would show a boundary at \(x = 0.5\). Place a solid line or marker at this point because the inequality includes \(x = \frac{1}{2}\) (thanks to the 'greater than or equal to' portion). Everything to the right of this boundary indicates numbers that satisfy the inequality condition.

To confirm, you may use a graphing calculation tool. Enterting the expression effectively visualizes the shaded region extending rightward from \(x = \frac{1}{2}\), verifying that all those points are valid solutions.

Number Line Representation

Using a number line to represent solutions of an inequality is intuitive and straightforward. For \(x \geq \frac{1}{2}\), the number line graphically highlights every solution in a clear manner. Mark \(\frac{1}{2}\) clearly on the line. Since the inequality is "greater than or equal to," a filled dot or circle is placed on \(\frac{1}{2}\) to show it is included in the solution set.

To illustrate all possible solutions, draw an arrow starting from \(\frac{1}{2}\) and moving to the right indefinitely. This arrow indicates every point beyond \(\frac{1}{2}\) also satisfies the inequality.

Number line representations are particularly helpful because they provide a simple, visual representation of where values fall in relation to the constraint set by the inequality.