Problem 88 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each inequality. $$5(x+4)>5 x+10$$

Step-by-Step Solution

Answer

The solution to the inequality \(5(x+4)>5x+10\) is all real numbers.

Step 1: Distribute

Distribute the 5 in \(5(x+4)\) to get \(5x + 20 > 5x + 10.\)

Step 2: Simplify the inequality

Subtract \(5x\) from both sides of the inequality to get \(20 > 10\)

Step 3: Interpret the Result

The inequality \(20 > 10\) is always true, meaning that for every value of \(x\), the original inequality is also true. Therefore, the solution is all real numbers.

Key Concepts

Distributive Property

The distributive property is a useful algebraic tool that allows us to multiply a single term across terms within parentheses. This property simplifies expressions and makes equations easier to solve.
In the provided exercise, you are given an inequality in the form of \(5(x+4) > 5x+10\). To apply the distributive property, we multiply the number 5 by each term inside the parentheses \((x+4)\):

• Distribute 5 to \(x\): \(5 \cdot x = 5x\)
• Distribute 5 to 4: \(5 \cdot 4 = 20\)

This gives us a new expression \(5x + 20\) on the left side, converting the inequality to \(5x + 20 > 5x + 10\).
Applying the distributive property is the first step in solving the given inequality and helps simplify the expression for further steps.

Simplifying Inequalities

Once you've used the distributive property, the next crucial step is simplifying the inequality further. Our goal is to isolate the variable or simplify the inequality as much as possible. In our inequality \(5x + 20 > 5x + 10\), we notice that \(5x\) appears on both sides.
To simplify this, we subtract \(5x\) from both sides to remove it:

• Subtract \(5x\) from the left: \(5x + 20 - 5x = 20\)
• Subtract \(5x\) from the right: \(5x + 10 - 5x = 10\)

This simplification leads us to the inequality \(20 > 10\), a simple statement that we know to be true. By simplifying, you reveal clearer insights into the problem, leading us to the correct solution.

All Real Numbers as Solution

Inequalities can sometimes lead us to interesting results. In this case, after simplification, where we obtained \(20 > 10\), we have a statement that is always true, no matter what value \(x\) takes.
This implies that the inequality does not depend on the variable \(x\) in any meaningful way within the given problem. Since \(20 > 10\) holds for all values of \(x\), the solution set for this inequality is all real numbers.
When an inequality results in a statement that is universally true (or false), it reflects on the nature of the equation:

• If it is always true, then every value of \(x\) satisfies the inequality.
• If it is always false, then no value of \(x\) can satisfy it.

In this scenario, because our inequality resolved to a constantly true statement, we conclude that every real number is indeed a solution to the original inequality.