Problem 45 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(8\left(\frac{1}{2} x+3\right)<8\left(\frac{1}{2} x-1\right)\)

Step-by-Step Solution

Answer

No solutions. The empty set \(\text{∅}\).

Step 1 - Distribute 8 on both sides

Distribute the 8 in the given inequality to remove the parenthesis on both sides.Equation: \[8 \times \frac{1}{2}x + 8 \times 3 < 8 \times \frac{1}{2}x - 8\text{.}\]Simplify each term:\[4x + 24 < 4x - 8\text{.}\]

Step 2 - Isolate terms with variables

Subtract \(4x\) from both sides to isolate the constants.\[4x + 24 - 4x < 4x - 8 - 4x\]Simplify the equation:\[24 < -8\text{.}\]

Step 3 - Evaluate the inequality

Since \[24 < -8\] is a false statement, there are no solutions to the inequality.

Step 4 - Graph the solution set

Because there are no solutions, the graph will be empty. There are no values of \(x\) that satisfy the inequality.

Step 5 - Write in interval notation

In interval notation, the solution set is written as the empty set \(\text{∅}\).

Key Concepts

Distributive Property in Algebra

The distributive property is a fundamental rule in algebra used to simplify expressions and solve equations. It states that multiplying a sum by a number gives the same result as multiplying each addend and then adding the products. For example, if we have an expression like

8(\( \frac{1}{2} x + 3 \))

, we apply the distributive property as follows:

• Multiply 8 by \( \frac{1}{2} x \) to get 4x.
• Multiply 8 by 3 to get 24.

That means the expression inside the parentheses is removed:
4x + 24
The same rule applies on the right-hand side, turning

8(\( \frac{1}{2} x - 1 \))

into

4x - 8.
This way, we easily transform the inequality and move to the next step.

Isolating Variables in Inequalities

Isolating the variable is essential to solve inequalities and understand what values work for the inequality. For instance, in our problem:

4x + 24 < 4x - 8

we need to move all terms involving x to one side. In this case, we subtract 4x from both sides:

• On the left, you get: 4x + 24 - 4x, which simplifies to 24.
• On the right, you get: 4x - 8 - 4x, which simplifies to -8.

So, after isolating terms involving x, we have:

24 < -8.
Realizing this inequality is false is key. It means there's no value of x that makes 24 less than -8. No solutions mean we conclude our process.

Interval Notation

Interval notation is a way of writing sets of solutions, mainly used to express the range of values. Since we determined the solution has no values of x, it implies we have an empty set. An empty set can be represented in interval notation as:

• ∅ (empty set symbol)
• ( ) (parentheses showing no numbers within a certain range)

Using interval notation helps quickly identify solution ranges or specify no solution exists. It's especially useful in more complex inequalities and when graphing the solution sets on a number line.