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Chapter 4: Problem 18

Solve each inequality. $$5(e-2)>10$$

Short Answer

Expert verified
e > 4

Step by step solution

01

- Distribute the 5

First, apply the distributive property to multiply 5 with the terms inside the parentheses: \[5(e-2)>10\]\[5e - 10 > 10\]
02

- Isolate the term with the variable

Add 10 to both sides of the inequality to isolate the term with the variable: \[5e - 10 + 10 > 10 + 10\]\[5e > 20\]
03

- Solve for the variable

Divide both sides of the inequality by 5 to solve for the variable: \[\frac{5e}{5} > \frac{20}{5}\]\[e > 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra. It allows us to simplify expressions and solve equations more easily. When you see an expression like \(5(e-2)\), the distributive property tells us to multiply each term inside the parentheses by the number outside. So, we multiply 5 by \(e\) and 5 by \(-2\), which gives us:

\[5 \times e - 5 \times 2 = 5e - 10\]
This simplification turns our problem into a more straightforward form, helping us proceed to the next steps, like isolating variables. Understanding and applying the distributive property is essential for solving many types of algebraic equations and inequalities.
Isolating Variables
After using the distributive property, the next step is isolating the variable. This is crucial because it allows you to solve for the variable. In the inequality \(5e - 10 > 10\), we need to get \(e\) by itself on one side of the inequality.
To do this, we can add 10 to both sides:

\[5e - 10 + 10 > 10 + 10\]
This simplifies to:

\[5e > 20\]
Now, the variable \(e\) is isolated on one side, making it much easier to solve the inequality in the next step. Keep in mind that whatever operation you perform on one side of the inequality, you must do to the other side to maintain balance.
Solving Inequalities
The final step in solving the inequality \(5e > 20\) is to completely solve for \(e\). This requires you to perform one last operation: dividing both sides by 5.

When you divide both sides by 5, you get:

\[ \frac{5e}{5} > \frac{20}{5} \]
Which simplifies to:

\[ e > 4 \]
So, the solution to the inequality \(5(e-2)>10\) is \(e > 4\).
It's important to understand that inequalities indicate a range of possible solutions. In this case, any value greater than 4 for \(e\) will satisfy the inequality. Always double-check your steps to ensure accuracy, especially when dealing with inequalities.

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