Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(6 x-4 \leq 2+8 x\)

Step-by-Step Solution

Answer

The solution to the inequality \(6x - 4 \leq 2 + 8x\) is \(x \geq -3\). The graph is a ray that starts at -3 and extends to the right, with a filled circle at -3.

Step 1: Simplify the Inequality

Subtract 6x from both sides, the inequality becomes \(-4 \leq 2+8x - 6x\)

Step 2: Solve for x

Continue to simplify the inequality to get the variable x alone on one side. Simplify the right side to get \(-4 \leq 2+2x\). Then subtract 2 from both sides to get \(-6 \leq 2x\). Finally, divide both sides by 2 to solve for x. This gives \(x \geq -3\) as a solution set.

Step 3: Graph the solution on the real number line

The graph is a ray that starts at -3 and extends to the right. The starting point at -3 is represented by a filled circle because the inequality is \(x \geq -3\), which includes -3 in the solution set.

Key Concepts

Real Number Line

The real number line is a fundamental concept in mathematics that helps to visualize the set of real numbers in a linear fashion. Imagine a straight, horizontal line stretching infinitely in both directions. On this number line, each point corresponds to a real number, arranged from smallest to largest as you move from left to right.
The negative numbers are located to the left of zero, while positive numbers sit to the right. Zero sits in the middle, acting as a sort of hinge or pivot point.

• Negative numbers appear to the left.
• Positive numbers are found on the right.
• Fractions and irrational numbers can be located between integers.

In problems involving inequalities, the real number line is especially useful for representing solution sets. Once you solve an inequality, such as determining that \(x \geq -3\), you can use the number line to show all possible solutions visually. In this case, a filled circle at -3 indicates that -3 itself is included in the solution set, and a line extending to the right indicates that all numbers greater than -3 are also solutions. This straightforward graphical approach makes interpreting inequality results much clearer.

Graphical Representation

Graphical representation involves depicting mathematical ideas and solutions visually. When it comes to inequalities, representation on the real number line is a powerful tool. Once you've solved an inequality, drawing it on a number line can significantly clarify what the solution means.The graph for the inequality \(x \geq -3\) uses a ray to indicate solutions:

• A filled circle at -3 signifies that -3 is part of the solution.
• We extend the line, or ray, to the right, which shows that every number larger than -3 is also a solution.

This method offers an immediate, visual understanding of which values satisfy the inequality. Students often find these visual aids very helpful, as they turn abstract concepts into something much more tangible. The strength in graphical representations lies in their ability to provide clarity and intuitive comprehension of the problem at hand.

Algebraic Manipulation

Algebraic manipulation is the process of restructuring equations or inequalities to isolate variables or find solutions. For the inequality \(6x - 4 \leq 2 + 8x\), we use algebraic manipulation to find the solution by simplifying and isolating \(x\).
Starting with this inequality, the goal is to get \(x\) by itself on one side of the inequality symbol. Here’s a breakdown:

• First, subtract \(6x\) from both sides to simplify the inequality to \(-4 \leq 2 + 2x\).
• Next, subtract 2 from both sides, leading to \(-6 \leq 2x\).
• Finally, divide both sides by 2 to isolate \(x\), resulting in \(x \geq -3\).

Each step involves keeping the balance of the inequality the same while performing the inverse operations. When dealing with inequalities, especially involving variables, this process ensures that the solution is both correct and easy to understand. Ensuring each step is clear will help in visualizing the effect each manipulation has on the inequality, aiding in mastering algebraic understanding.