(b) Substitute i $$ y=1 $$, ii $$ y=1.5 $$ and iii $$ y=2 $$ into $$ 4(2 y+3)-5 y

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(b) Substitute 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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To verify that the solution set for the inequality $$ 4(2y + 3) - 5y < 18 - y $$ is correct, we'll substitute the given values of $$ y $$ into the inequality and check whether the inequality holds true.

### Step 1: Simplify the Inequality
Before substituting, it's helpful to simplify the inequality:

$$
\begin{align*}
4(2y + 3) - 5y &< 18 - y \
8y + 12 - 5y &< 18 - y \
3y + 12 &< 18 - y \
3y + y + 12 &< 18 \quad (\text{Add } y \text{ to both sides}) \
4y + 12 &< 18 \
4y &< 6 \quad (\text{Subtract } 12 \text{ from both sides}) \
y &< 1.5 \quad (\text{Divide both sides by } 4)
\end{align*}
$$

**Solution Set:** $$ y < 1.5 $$

### Step 2: Substitute the Given Values

Let's substitute each value of $$ y $$ into the original inequality to verify whether it satisfies $$ y < 1.5 $$.

#### i. Substitute $$ y = 1 $$

$$
\begin{align*}
4(2(1) + 3) - 5(1) &< 18 - 1 \
4(2 + 3) - 5 &< 17 \
4(5) - 5 &< 17 \
20 - 5 &< 17 \
15 &< 17 \quad \text{✅ True}
\end{align*}
$$

**Conclusion:** $$ y = 1 $$ satisfies the inequality.

#### ii. Substitute $$ y = 1.5 $$

$$
\begin{align*}
4(2(1.5) + 3) - 5(1.5) &< 18 - 1.5 \
4(3 + 3) - 7.5 &< 16.5 \
4(6) - 7.5 &< 16.5 \
24 - 7.5 &< 16.5 \
16.5 &< 16.5 \quad \text{❌ False}
\end{align*}
$$

**Conclusion:** $$ y = 1.5 $$ does **not** satisfy the inequality since $$ 16.5 $$ is not less than $$ 16.5 $$.

#### iii. Substitute $$ y = 2 $$

$$
\begin{align*}
4(2(2) + 3) - 5(2) &< 18 - 2 \
4(4 + 3) - 10 &< 16 \
4(7) - 10 &< 16 \
28 - 10 &< 16 \
18 &< 16 \quad \text{❌ False}
\end{align*}
$$

**Conclusion:** $$ y = 2 $$ does **not** satisfy the inequality.

### Step 3: Summary

- **$$ y = 1 $$:** Satisfies $$ y < 1.5 $$ (True)
- **$$ y = 1.5 $$:** Does **not** satisfy $$ y < 1.5 $$ (Boundary case, False)
- **$$ y = 2 $$:** Does **not** satisfy $$ y < 1.5 $$ (False)

These substitutions confirm that the solution set $$ y < 1.5 $$ is correct.
When $$ y = 1 $$, the inequality holds true. However, when $$ y = 1.5 $$ and $$ y = 2 $$, the inequality does not hold. This confirms that the solution set is $$ y < 1.5 $$.

(b) Substitute 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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