Question

writing an inequality to include a solution
ms. cassidy plotted the point (2, 3) on miguel’s graph of ( y < 2x - 4 ). she instructed him to change one number or one symbol in his inequality so that the point (2, 3) can be included in the solution set.
which equations might miguel write? check all that apply.
( y < 2x - 1 )
( y < 2x - 4 )
( y > 2x - 4 )
( y < 2x + 4 )
( y < 3.5x - 4 )
( y < 4x - 4 )

Explanation:

Step1: Test point (2,3) on each inequality

For $y < 2x - 1$:
Substitute $x=2, y=3$: $3 < 2(2)-1 = 3$ → $3 < 3$ is false.

For $y \leq 2x - 4$:
Substitute $x=2, y=3$: $3 \leq 2(2)-4 = 0$ → $3 \leq 0$ is false.

For $y > 2x - 4$:
Substitute $x=2, y=3$: $3 > 2(2)-4 = 0$ → $3 > 0$ is true.

For $y < 2x + 4$:
Substitute $x=2, y=3$: $3 < 2(2)+4 = 8$ → $3 < 8$ is true.

For $y < 3.5x - 4$:
Substitute $x=2, y=3$: $3 < 3.5(2)-4 = 3$ → $3 < 3$ is false.

For $y < 4x - 4$:
Substitute $x=2, y=3$: $3 < 4(2)-4 = 4$ → $3 < 4$ is true.

Step2: Verify single modification rule

Original inequality: $y < 2x - 4$

• $y > 2x - 4$: Changed symbol ($<$ to $>$) → 1 change.
• $y < 2x + 4$: Changed constant ($-4$ to $+4$) → 1 change.
• $y < 4x - 4$: Changed coefficient ($2$ to $4$) → 1 change.

Answer:

$y > 2x - 4$
$y < 2x + 4$
$y < 4x - 4$