(8) Show that the solution set to the inequality $$ 4(2 y+3)-5 y

Question

(8) Show that the solution set to the inequality $$ 4(2 y+3)-5 y<18-y $$ is $$ y<1.5 $$ (bubstitute $$ \mathbf{i} y=1 $$, ii $$ y=1.5 $$ and iii $$ y=2 $$ into $$ 4(2 y+3)-5 y<18-y $$ to show that the solution set is correct.

UpStudy AI · Accepted Answer

To solve the inequality $$ 4(2y + 3) - 5y < 18 - y $$ and verify that the solution set is $$ y < 1.5 $$, follow these steps:

### **Part (a): Solving the Inequality**

1. **Expand the Left Side:**
   $$
   4(2y + 3) - 5y = 8y + 12 - 5y
   $$
   
2. **Combine Like Terms:**
   $$
   8y + 12 - 5y = 3y + 12
   $$
   
3. **Set Up the Inequality:**
   $$
   3y + 12 < 18 - y
   $$
   
4. **Move All Terms involving $$ y $$ to the Left Side and Constants to the Right Side:**
   $$
   3y + y < 18 - 12
   $$
   
5. **Simplify:**
   $$
   4y < 6
   $$
   
6. **Solve for $$ y $$:**
   $$
   y < \frac{6}{4} \implies y < 1.5
   $$
   
   Hence, the solution set is $$ y < 1.5 $$.

### **Part (b): Verifying the Solution**

Substitute the values $$ y = 1 $$, $$ y = 1.5 $$, and $$ y = 2 $$ into the original inequality to verify the solution.

1. **Substitute $$ y = 1 $$:**
   $$
   4(2(1) + 3) - 5(1) < 18 - 1
   $$
   $$
   4(2 + 3) - 5 < 17
   $$
   $$
   4(5) - 5 < 17
   $$
   $$
   20 - 5 < 17
   $$
   $$
   15 < 17 \quad \text{(True)}
   $$
   
2. **Substitute $$ y = 1.5 $$:**
   $$
   4(2(1.5) + 3) - 5(1.5) < 18 - 1.5
   $$
   $$
   4(3 + 3) - 7.5 < 16.5
   $$
   $$
   4(6) - 7.5 < 16.5
   $$
   $$
   24 - 7.5 < 16.5
   $$
   $$
   16.5 < 16.5 \quad \text{(False)}
   $$
   
3. **Substitute $$ y = 2 $$:**
   $$
   4(2(2) + 3) - 5(2) < 18 - 2
   $$
   $$
   4(4 + 3) - 10 < 16
   $$
   $$
   4(7) - 10 < 16
   $$
   $$
   28 - 10 < 16
   $$
   $$
   18 < 16 \quad \text{(False)}
   $$
   
**Conclusion:**
- For $$ y = 1 $$, the inequality holds true.
- For $$ y = 1.5 $$ and $$ y = 2 $$, the inequality does not hold.
  
This confirms that the solution set $$ y < 1.5 $$ is correct.

#### **Graphical Representation:**

To further visualize the solution, consider a number line:

$$
\text{---|-----|-----|-----|-----|-----}
\quad 1 \quad \quad 1.5 \quad \quad 2
$$

- The solution set includes all values **to the left** of $$ 1.5 $$.

---

**Final Answer:**  
After simplifying, all real numbers y with y < 1.5 satisfy the inequality. Substituting y = 1 makes 15 < 17 true, y = 1.5 makes 16.5 < 16.5 false, and y = 2 makes 18 < 16 false. Thus, the solution set is y < 1.5.
The solution set for the inequality $$ 4(2y + 3) - 5y < 18 - y $$ is $$ y < 1.5 $$.

(8) Show that the solution set to the inequality \( 4(2 y+3)-5 y<18-y \) is \( y<1.5 \) (bubstitute \( \mathbf{i} y=1 \), ii \( y=1.5 \) and iii \( y=2 \) into \( 4(2 y+3)-5 y<18-y \) to show that the solution set is correct.

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