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Chapter 3: Problem 41

Solve each inequality. $$4(w+2)-3(w-1)>5(w-1)-5 w$$

Short Answer

Expert verified
w > -16.

Step by step solution

01

- Distribute the terms

Distribute the terms on both sides of the inequality: \[4(w+2)-3(w-1)>5(w-1)-5w\]. This simplifies to: \[4w + 8 - 3w + 3 > 5w - 5 - 5w\].
02

- Combine like terms

Combine like terms on each side: \[(4w - 3w) + (8 + 3) > (5w - 5w) - 5\]. This simplifies to: \[w + 11 > -5\].
03

- Isolate the variable

Subtract 11 from both sides of the inequality to isolate the variable \(w\): \[w + 11 - 11 > -5 - 11\]. This simplifies to: \[w > -16\].

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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 used to remove parentheses by distributing a multiplication operation over an addition or subtraction inside the parentheses.
In the exercise, we start with the inequality \ \[4(w + 2) - 3(w - 1) > 5(w - 1) - 5w. \]
Apply the distributive property as follows:
* First, distribute 4 across \(w + 2\), giving \(4w + 8\).
* Next, distribute -3 across \(w - 1\), giving \(-3w + 3\).
* On the right side, distribute 5 across \(w - 1\), giving \(5w - 5\). Note that \(-5w\) remains as is.
This results in:
\[4w + 8 - 3w + 3 > 5w - 5 - 5w.\]
Understanding the distributive property helps simplify the expressions, making it easier to solve the inequality.
Combining Like Terms
Combining like terms means putting together terms that have the same variable(s) to simplify an expression.
In our exercise, after applying the distributive property, we get: \ \[4w + 8 - 3w + 3 > 5w - 5 - 5w \]
To combine like terms, follow these steps:
* Group the terms with \(w\) together: \(4w - 3w\).
* Group the constants together: \(8 + 3\).
* Simplify the right side: since \(5w\) and \(-5w\) cancel each other out, it simplifies to \(-5\).
This leads to:
\ \[4w - 3w + 8 + 3 > -5.\]
Simplify further to get: \ \[w + 11 > -5.\]
Combining like terms is crucial for simplifying expressions quickly and accurately.
Isolate the Variable
Isolating the variable involves performing operations to get the variable by itself on one side of the equation or inequality.
In our simplified inequality, we have: \ \[w + 11 > -5. \]
To isolate \(w\), subtract 11 from both sides:
\ \[w + 11 - 11 > -5 - 11.\]
This simplifies to:
\ \[w > -16.\]
Now, \(w\) is isolated, and the solution is clear. It states that \(w\) must be greater than \(-16\).
Isolating the variable is a critical step in solving any equation or inequality, as it directly leads to finding the solution.

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