ework: HW2 Spring 2025 Solve the linear inequality. Express the solution set using set-builder and interval notation. Graph the solution set. $$ 8(x+4)-3 x \leq 8(x-1)+19 $$ The solution set expressed in set-builder notation is $$ \{x \mid 1\} $$

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UpStudy AI · Accepted Answer

Let's solve the linear inequality step by step.

1. **Start with the given inequality:**
   $$
   8(x+4) - 3x \leq 8(x-1) + 19
   $$

2. **Distribute the terms:**
   - On the left side: 
     $$
     8(x+4) = 8x + 32
     $$
     So, the left side becomes:
     $$
     8x + 32 - 3x = 5x + 32
     $$
   - On the right side:
     $$
     8(x-1) = 8x - 8
     $$
     So, the right side becomes:
     $$
     8x - 8 + 19 = 8x + 11
     $$

3. **Rewrite the inequality:**
   $$
   5x + 32 \leq 8x + 11
   $$

4. **Isolate $$x$$:**
   - Subtract $$5x$$ from both sides:
     $$
     32 \leq 3x + 11
     $$
   - Subtract $$11$$ from both sides:
     $$
     21 \leq 3x
     $$
   - Divide by $$3$$:
     $$
     7 \leq x
     $$
   - This can also be written as:
     $$
     x \geq 7
     $$

5. **Express the solution set:**
   - In set-builder notation, the solution set is:
     $$
     \{x \mid x \geq 7\}
     $$
   - In interval notation, the solution set is:
     $$
     [7, \infty)
     $$

6. **Graph the solution set:**
   - The graph will show a closed circle at $$7$$ and a line extending to the right towards infinity.

Now, let's summarize the solution:

- The solution set expressed in set-builder notation is $$ \{x \mid x \geq 7\} $$.
- The solution set expressed in interval notation is $$ [7, \infty) $$.

Next, I will graph the solution set.

Since I cannot create visual graphs directly, you can visualize it as follows:
- Draw a number line.
- Place a closed circle at $$7$$ to indicate that $$7$$ is included in the solution.
- Shade the line to the right of $$7$$ to indicate all numbers greater than $$7$$ are included in the solution set.
The solution set is all real numbers $$ x $$ such that $$ x \geq 7 $$. In set-builder notation, this is $$ \{x \mid x \geq 7\} $$, and in interval notation, it's $$ [7, \infty) $$.

ework: HW2 Spring 2025 Solve the linear inequality. Express the solution set using set-builder and interval notation. Graph the solution set. \[ 8(x+4)-3 x \leq 8(x-1)+19 \] The solution set expressed in set-builder notation is \( \{x \mid 1\} \)

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