Solve the linear inequality. Other than empty set​, graph the solution set on a number line.
3 left parenthesis x plus 1 right parenthesis less than or equals 2 left parenthesis x plus 6 right parenthesis plus x

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set in interval notation is 
  
enter your response here.
​(Simplify your answer. Use integers or fractions for any numbers in the expression. Type your answer in interval​ notation.)
B.
The solution set is empty set.

We are given the inequality:

\[
3(x + 1) \leq 2(x + 6) + x
\]

Let's solve this step by step.

### Step 1: Expand both sides
Distribute the constants on both sides of the inequality:

\[
3(x + 1) = 3x + 3
\]
\[
2(x + 6) = 2x + 12
\]

Now substitute these into the inequality:

\[
3x + 3 \leq 2x + 12 + x
\]

### Step 2: Combine like terms
Simplify the right-hand side by combining the \(2x\) and \(x\):

\[
3x + 3 \leq 3x + 12
\]

### Step 3: Isolate the variable
Subtract \(3x\) from both sides:

\[
3 \leq 12
\]

### Step 4: Interpret the result
The inequality \(3 \leq 12\) is always true. This means that the inequality holds for all values of \(x\).

### Conclusion
The solution set is all real numbers.

### Step 5: Express the solution in interval notation
In interval notation, this is represented as:

\[
(-\infty, \infty)
\]

Thus, the correct choice is:

**A.** The solution set in interval notation is \((- \infty, \infty)\).

### Graphing the solution set
To graph the solution set on a number line, shade the entire number line because all real numbers are solutions.

---

Would you like any further clarification on this? Here are 5 related questions to expand on this topic:

1. How do you solve inequalities with absolute values?
2. What is the difference between strict inequalities and non-strict inequalities?
3. How do you solve compound inequalities?
4. Can an inequality have more than one solution set?
5. How do you graph inequalities on the coordinate plane?

**Tip:** When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Solve the linear inequality. Other than empty set, graph the solution set on a number line. 3(x + 1) ≤ 2(x + 6) + x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Learn how to solve the linear inequality 3(x + 1) ≤ 2(x + 6) + x with a step-by-step guide and graph the solution set on a number line. This problem covers key algebraic concepts, including the distributive property and properties of inequalities. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation