Problem 43 Graph the solution set, and writ... [FREE SOLUTION]

Chapter 8: Problem 43

Graph the solution set, and write it using interval notation. $$ 3(2 x-4)-4 x<2 x+3 $$

Short Answer

Expert verified

The solution set is all real numbers, which can be written in interval notation as (- 鈭�, 鈭�).

Step by step solution

Distribute the multiplication

Distribute the 3 through the parentheses on the left side of the inequality: 3(2x - 4) - 4x < 2x + 3 6x - 12 - 4x < 2x + 3

Combine like terms

Combine the like terms on the left side of the inequality: 6x - 4x - 12 < 2x + 3 2x - 12 < 2x + 3

Subtract 2x from both sides

Subtract 2x from both sides to get rid of the variable term on the right side: 2x - 12 - 2x < 2x + 3 - 2x -12 < 3

Conclusion

Since the inequality -12 < 3 is always true, the inequality is true for all values of x. Therefore, the solution set is all real numbers.

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Key Concepts

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

graphing solution sets

Graphing solution sets can help you visualize inequalities. To graph the solution of an inequality on a number line, follow these steps:
First, determine whether the inequality includes equal to (鈮� or 鈮�) or not (< or >). This will guide you on whether to use an open dot (for < and >) or a closed dot (for 鈮� and 鈮�).
In our case, the inequality is always true -12 < 3, making the solution set all real numbers. Therefore, we shade the entire number line to represent this.
Remember, shading the entire number line indicates that every number is a solution.

interval notation

Interval notation offers a compact way to represent solution sets. It uses brackets and parenthesis to denote if end values are included or excluded.
For the solution that includes all real numbers, we write it using interval notation as $(-鈭�, 鈭�)$.
Here's a quick guide:

Use parenthesis ( ) for values that are not included.
Use brackets [ ] for values that are included.

combining like terms

Combining like terms is essential in simplifying expressions. Like terms have the same variable raised to the same power.
In our problem:

6x and -4x are like terms.
Combine them to get 2x.

This practice reduces complexity and makes it easier to solve inequalities.
Always look for and combine like terms early in a problem.

distributive property

The distributive property simplifies expressions by distributing multiplication over addition or subtraction.
In our problem, we use it to get rid of parentheses: $3(2x - 4)\rightarrow 6x - 12$.
This property states that $a(b+c)=ab+ac$.
It helps in breaking down complex expressions, making them easier to manage.

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91影视

Graph the solution set, and write it using interval notation. $$ 3(2 x-4)-4 x<2 x+3 $$

Short Answer

Step by step solution

Distribute the multiplication

Combine like terms

Subtract 2x from both sides

Conclusion

Key Concepts

graphing solution sets

interval notation

combining like terms

distributive property

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