Problem 65 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{1}{2} x+6 \geq 4+2 x $$

Step-by-Step Solution

Answer

The solution is \( x \leq \frac{4}{3} \), or \((-\infty, \frac{4}{3}]\).

Step 1: Move Variables to One Side

Subtract \(2x\) from both sides of the inequality to get the variables on one side. This gives: \[ \frac{1}{2}x + 6 - 2x \geq 4 \] which simplifies to: \[ -\frac{3}{2}x + 6 \geq 4 \]

Step 2: Isolate the Variable Term

Subtract 6 from both sides to isolate the term with the variable. This gives: \[ -\frac{3}{2}x \geq -2 \]

Step 3: Solve for x

Divide both sides by \(-\frac{3}{2}\). Remember that dividing by a negative number reverses the inequality sign:\[ x \leq \frac{4}{3} \]

Step 4: Write Solution in Interval Notation

The inequality \(x \leq \frac{4}{3}\) is expressed in interval notation as: \[ (-\infty, \frac{4}{3}] \]

Step 5: Graph the Solution Set

On a number line, plot a point at \(\frac{4}{3}\) and shade all numbers to the left. Use a closed dot at \(\frac{4}{3}\) to indicate that the endpoint is included.

Key Concepts

Interval Notation

Interval notation is a way of expressing the set of solutions to an inequality. It offers a concise method to describe which numbers satisfy a given condition.
For example, when we have an inequality like \( x \leq \frac{4}{3} \), interval notation helps us represent the range of numbers that fulfill this condition. What we do is recognize that \( x \) can be any number less than or equal to \( \frac{4}{3} \).

To express this using interval notation, we write:

• \( (-\infty, \frac{4}{3}] \)

The interval starts at negative infinity and goes up to \( \frac{4}{3} \), including \( \frac{4}{3} \) itself. Note the use of a parenthesis to denote that infinity is not a number that can be reached, hence not included. The square bracket at \( \frac{4}{3} \) indicates that this endpoint is part of the solution.
Understanding interval notation is vital as it clearly communicates which parts of the number line are included in the solution set.

Graphing Solution Sets

Graphing solution sets of inequalities is an essential technique to visualize the range of solutions. When we say we need to graph an inequality like \( x \leq \frac{4}{3} \), we're essentially illustrating all the values of \( x \) that satisfy this condition on a number line.

To graph:

• First, locate the number \( \frac{4}{3} \) on the number line.
• Since our inequality allows for \( x \) to be equal to \( \frac{4}{3} \) as well, we place a closed dot at this point, indicating inclusion.
• Then, shade all the numbers to the left of \( \frac{4}{3} \) to negative infinity, showing all the values of \( x \) that make the inequality true.

This shaded region effectively demonstrates the solution set, offering a clear visual guide about where the solutions lie. Graphing is helpful because it provides a quick, intuitive understanding of the inequality's solutions.

Inequality Transformations

Solving inequalities often involves performing transformations to isolate the variable. These transformations must be done carefully due to the unique properties of inequalities.
Consider the inequality: \( \frac{1}{2}x + 6 \geq 4 + 2x \). Our goal is to solve for \( x \) by getting it alone on one side.Some key steps:

• - Move all terms involving \( x \) to one side by subtracting or adding appropriately. This could change the appearance of the inequality but maintains its truth.
• - When you multiply or divide by a negative number, be aware that the inequality sign must be flipped. For instance, when we divided by \(-\frac{3}{2}\), we reversed the \( \geq \) to \( \leq \).

These transformations are critical in maintaining the inequality's validity while simplifying it. Each step gets you closer to easily identifying the values that satisfy the inequality. Mastering this concept can make solving equations and inequalities less daunting.