Problem 23 - 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. $$\frac{3}{4} x-6 \leq x-7$$

Step-by-Step Solution

Answer

The solution to the inequality \(\frac{3}{4}x - 6 \leq x - 7\) is \(x \leq 4\). This can be represented on the number line as a solid circle at 4 with a line extending to the left, indicating that x is any number less than or equal to 4.

Step 1: Simplify the inequality

Start by subtracting \(\frac{3}{4}x\) from both sides of the inequality to solve for x: \[x-\frac{3}{4}x \leq 7-6\] This simplifies to: \[\frac{1}{4}x \leq 1\].

Step 2: Solve for x

Multiply both sides of the inequality by 4 to solve for x. Doing so, you get: \[x \leq 4 \]. That is the solution in the inequality notation.

Step 3: Draw the solution on the number line

Sketch a number line, label it with numbers 1 to 5. Draw a solid circle at 4 and draw a line towards the left ending with an arrow, for all the numbers less than or equal to 4. This represents all possible solutions for x.

Key Concepts

Solving Inequalities

To solve inequalities, you'll need to manipulate the inequality to isolate the variable, much like you solve equations. However, there are some important differences to keep in mind:

• When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.
• The inequality does not change if you add or subtract the same number from both sides.

In our original exercise, we started with the inequality \(\frac{3}{4} x - 6 \leq x - 7\). By subtracting \(\frac{3}{4}x\) from both sides, we simplified it to \(\frac{1}{4}x \leq 1\).
Then, multiplying both sides by 4, we obtained \(x \leq 4\). This means that \(x\) can be any value less than or equal to 4. Remember, always double-check your steps to ensure accuracy!

Graphing Inequalities

When you graph inequalities, you're visualizing a range of values that satisfy the inequality condition. Let's consider how you would graph \(x \leq 4\) on a number line.
On a Cartesian plane, you usually shade the region that represents all solutions to the inequality:

• If the inequality is \(\leq\) or \(\geq\), use a solid line to indicate that the points on the line are included in the solution.
• If it's \(<\) or \(>\), use a dashed line to indicate the boundary isn't included.

A number line representation is slightly different and will focus solely on one dimension but follows the same principles for including or not including boundary points.

Number Line Representation

A number line is a perfect tool for showing solutions to inequalities. It helps you visually grasp the set of valid solutions for an inequality. For \(x \leq 4\), you should:

• Draw a solid filled circle at 4 since \(4\) is included in the solution set.
• Draw a line extending to the left, covering all numbers less than 4.
• Finally, add an arrow at the end of the line to indicate that the solution extends indefinitely in that direction.

This visual is a simple and effective way to understand which numbers satisfy the inequality.

Solution Verification

Once you've solved an inequality and graphed it, it's always a wise step to verify your solution. You can use a graphing calculator or software to compare. If using a graphing utility:

• Enter the original inequality to visualize solutions.
• Ensure that the graph matches your paper-based solution. For \(x \leq 4\), you should confirm the solution starts at 4 and extends to the left on the x-axis.

Verification helps in catching any errors or misinterpretations in your manual calculations. You should see your shaded area or arrow accurately representing all possible values that satisfy the inequality.