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Chapter 3: Problem 26

Solve each inequality. $$2(w+4)-5 w<2$$

Short Answer

Expert verified
w > 2

Step by step solution

01

- Distribute the 2

Begin by distributing the 2 across the terms inside the parenthesis: 2(w+4) = 2w + 8. So, the inequality becomes 2w + 8 - 5w < 2.
02

- Combine Like Terms

Combine the terms involving w: 2w - 5w = -3w.So, now the inequality is -3w + 8 < 2.
03

- Isolate the w-term

Subtract 8 from both sides of the inequality to isolate the term with w: -3w + 8 - 8 < 2 - 8,which simplifies to: -3w < -6.
04

- Solve for w

Divide both sides of the inequality by -3, and remember to reverse the inequality sign because you are dividing by a negative number: \(-3w / -3 > -6 / -3\),which simplifies to: w > 2.

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Key Concepts

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

Distributing Terms in Algebra
When we distribute terms in algebra, we multiply a term outside the parenthesis by each term inside the parenthesis. In the provided exercise, we started with the expression \(2(w + 4)\). We need to distribute the 2 to both \(w\) and 4. This process is like sharing an equal amount of something with each item in a group. You multiply 2 by \(w\) and then 2 by 4:
\(2(w) = 2w\) and \(2(4) = 8\).
So, the expression becomes \(2w + 8\).
This step helps break down complex expressions into simpler ones, making them easier to work with in subsequent steps.
Combining Like Terms
Combining like terms means putting together terms that have the same variable raised to the same power. In our example, after distributing, we had the inequality: \(2w + 8 - 5w < 2\). Here, the like terms are \(2w\) and \(-5w\), both containing the variable \(w\).
To combine them:
  • Perform the operation: \(2w - 5w\)
  • This results in: \(-3w\)
Thus, our inequality simplifies to \(-3w + 8 < 2\). Combining like terms makes the expression simpler and helps in isolating the variable.
Isolating Variables
Isolating the variable means getting the term with the variable by itself on one side of the equation or inequality. This helps in finding the value of the variable. Continuing with our exercise example, we had \(-3w + 8 < 2\). To isolate \(w\):
Subtract 8 from both sides:
\(-3w + 8 - 8 < 2 - 8\)
Simplifying, we get:
\(-3w < -6\)
To solve for \(w\), divide both sides by \(-3\). Remember, dividing by a negative number reverses the inequality sign:
  • Divide: \(-3w / -3 > -6 / -3\)
  • This results in: \(w > 2\)
Therefore, \(w > 2\). Isolating the variable allows you to solve the inequality or equation.

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