Question

what is the error in solving and describing the graph of the inequality ( 3x - 2y < 8 - x )? what correction needs to be made?

| ( 3x - 2y < 8 - x ) |
| --- |
| step 1: ( -2y < 8 - 4x ) |
| step 2: ( y > -4 + 2x ) |
| step 3: ( y > 2x - 4 ) |
| step 4: solid line is graphed on ( y = 2x - 4 ) |
| step 5: test the point ( (0, 5) )
( y > 2x - 4 )
( 5 > 2(0) - 4 )
( 5 > -4 ) true |
| step 6: the region to the left/above the line is shaded. |

• step 3: ( y > 2x - 4 ) is a different inequality than ( y > -4 + 2x ), and it should be ( y > -2x + 4 )
• step 2: the inequality symbol should not have flipped; the inequality should be ( y < -4 + 2x )
• step 5: testing the point ( (0, 5) ) should look like this: ( 0 > 2(5) - 4 ), which is ( 0 > 6 ) and is false
• step 4: the line should be dashed because of the ( > ) symbol

Explanation:

First, re-solve the inequality $3x - 2y < 8 - x$ correctly to identify the error:

1. Combine like terms: $-2y < 8 - 4x$
2. Divide by -2 (flip inequality sign): $y > -4 + 2x$ (or $y > 2x - 4$)
3. For graphing: since the inequality is $>$ (not $\geq$), the boundary line $y=2x-4$ must be dashed.
4. Testing $(0,5)$: $5 > 2(0)-4$ simplifies to $5 > -4$, which is true, so the region above the line is correct.

Now evaluate the options:

• Step 3 is algebraically equivalent, so no error here.
• Step 2's sign flip was correct (dividing by negative), so this is wrong.
• Step 5's test calculation is misstated; the actual test is valid, so this is wrong.
• Step 4 uses a solid line, but $>$ requires a dashed line, which is the error.

Answer:

Step 4, the line should be dashed because of the > symbol