(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.

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.