Problem 29 Solve each inequality. $$4(y-3... [FREE SOLUTION]

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Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 29

Solve each inequality. $$4(y-3)-(3 y-12) \geq 2$$

Short Answer

Expert verified

y \geq 2

Step by step solution

- Distribute and combine like terms

Distribute the 4 into the parentheses \[4(y - 3) = 4y - 12\] and distribute the -1 into the parentheses \[-(3y - 12) = -3y + 12\]. Combine these results: \[4y - 12 - 3y + 12 \geq 2\].

- Simplify the expression

Combine the like terms: \[4y - 3y - 12 + 12 \geq 2\]. Which simplifies to \[y \geq 2\].

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Key Concepts

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

Distribution

When we solve inequalities or equations, we often need to use the distributive property.
The distributive property helps us to remove parentheses by distributing/multiplying a number outside the parenthesis to each term inside.
For example, in the inequality, we see \[4(y - 3)\].
Using the distributive property, we multiply 4 by both $y$ and $-3$:
\[4(y - 3) = 4y - 12\].
Similarly, we handle \[-(3y - 12)\], paying attention to the negative sign:
\[-(3y - 12) = -3y + 12\].
An easy way to remember this is to apply the number outside to each term within the parentheses step-by-step.

Combining Like Terms

Combining like terms is another essential skill when working with inequalities.
Here's how we combine like terms:

Identify terms that have the same variable.
Combine their coefficients.

For instance, after distributing in our exercise, we have \[-12\] and \[+12\] (terms without variables):

Combine \[-12 + 12\] to get 0.
This means these terms cancel out.

Now, focus on the terms with the variable \['y'\]: \[4y - 3y\]
The like terms here are \[4y\] and \[-3y\]. Combine them by subtracting the coefficients:
\[4y - 3y = y\].
The process of recognizing and combining these terms allows us to simplify the expression efficiently.

Simplifying Expressions

Simplifying expressions puts all our previously discussed skills to use.
First, make sure you distribute properly.
Once you鈥檝e combined like terms, what's left is to simplify further if necessary.
In the exercise, distributing and combining gives us:
\[-12 + 12\] cancels out;
\[-3y + 4y = y\].
The expression simplifies to:
\[y \geq 2\] Here's what to remember:

First, distribute.
Then, combine like terms.
Simplify by performing any remaining operations.

This systematic approach will guide you to the answer step-by-step, making it easier to manage even more complex inequalities.

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91影视

Solve each inequality. $$4(y-3)-(3 y-12) \geq 2$$

Short Answer

Step by step solution

- Distribute and combine like terms

- Simplify the expression

Key Concepts

Distribution

Combining Like Terms

Simplifying Expressions

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