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Chapter 3: Problem 34

Solve each inequality. $$5(x+1)-(5 x-1)>12$$

Short Answer

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Step by step solution

01

Distribute

Distribute the 5 on the left side: 5(x+1) = 5x + 5.
02

Rewrite the Inequality

Rewrite the inequality: 5x + 5 - (5x - 1) > 12.
03

Simplify Inside the Parentheses

Simplify inside the parentheses: 5x + 5 - 5x + 1 > 12.
04

Combine Like Terms

Combine like terms: 6 > 12.
05

Analyze the Result

Notice that the statement '6 > 12' is false. Therefore, there are no solutions for this inequality.

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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 math concept used to multiply a single term across terms inside parentheses. It states that for any numbers or variables, \( a(b + c) = ab + ac \).
In this exercise, we used the distributive property to expand the expression \( 5(x + 1) \).
Here's how we applied it:
\( 5(x + 1) = 5 \cdot x + 5 \cdot 1 = 5x + 5 \).
Remember, this property helps us by removing the parentheses, making the expression easier to work with.
simplifying expressions
Simplifying expressions involves performing arithmetic operations and combining like terms to make the expression as straightforward as possible.
In our exercise, after distributing, the expression was: \( 5x + 5 - (5x - 1) \).
To simplify inside the parentheses, first recognize the negative sign distributes as well:
\( 5x + 5 - 5x + 1 \).
Now, we can easily see the next step of the process, which brings us to combining like terms.
combining like terms
Combining like terms means to add or subtract terms that have the same variable raised to the same power.
In our simplified expression, \( 5x + 5 - 5x + 1 \), we can see that the \( 5x \) and \( -5x \) terms cancel each other out. What we are left with is: \[ 5 + 1 > 12 \]. This yields, \ 6 > 12 \.
This statement is false; therefore, there are no solutions for this inequality.
Combining like terms helps us to simplify our inequalities or equations to reach a clear and concise result.

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