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Chapter 2: Problem 68

Solve inequality and graph the solution set. \(2 x+3>2(x-4)\)

Short Answer

Expert verified
All real numbers, i.e., x 鈭� 鈩�.

Step by step solution

01

- Distribute and Simplify

Distribute the 2 on the right side of the inequality: 2(x - 4) = 2x - 8 So the inequality becomes: 2x + 3 > 2x - 8.
02

- Combine Like Terms

Subtract 2x from both sides of the inequality to combine like terms: 2x + 3 - 2x > 2x - 8 - 2x This simplifies to: 3 > -8.
03

- Interpret the Inequality

Since the inequality 3 > -8 is always true, it indicates that the original inequality is true for all values of x.
04

- Graph the Solution Set

The solution set is all real numbers because the inequality holds true for any x. On a number line, this is represented by shading the entire number line.

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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 key concept in algebra that helps us simplify expressions by distributing a multiplied term over addition or subtraction inside parenthesis. For example, in the expression 2(x - 4), you can use the distributive property to simplify it to 2x - 8.
This involves multiplying each term inside the parenthesis by 2:
\[ 2(x - 4) = 2 \times x - 2 \times 4 = 2x - 8 \] This property is essential when solving inequalities, as it ensures all terms are properly accounted for.
combining like terms
Combining like terms is a fundamental process in algebra that simplifies expressions by combining terms that have the same variable. For instance, in the expression 2x + 3 > 2x - 8, you can combine the terms with x:
Subtract 2x from both sides:
\[ 2x + 3 - 2x > 2x - 8 - 2x \] This simplifies to:

\[ 3 > -8 \] Combining like terms allows you to see whether an inequality or equation can be simplified further, as it strips away unnecessary terms for clearer insight.
graphing solutions
Graphing the solution of an inequality involves representing all possible solutions on a number line. For the inequality 2x + 3 > 2(x - 4), after simplifying, we find that 3 > -8, which is always true. This means the original inequality is true for all values of x.

On a number line, this solution is represented by shading the entire number line, indicating that all real numbers are solutions for x. Graphically showing solutions helps provide a visual understanding, making it easier to interpret complex inequalities and their solutions.

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