To solve the inequality \( 5(3 - x) < -2x + 6 \), we will start by distributing and simplifying both sides.
Step 1: Distribute on the left side
\[ 5(3 - x) = 15 - 5x \] So the inequality becomes: \[ 15 - 5x < -2x + 6 \]
Step 2: Add \( 5x \) to both sides
\[ 15 < -2x + 5x + 6 \] This simplifies to: \[ 15 < 3x + 6 \]
Step 3: Subtract 6 from both sides
\[ 15 - 6 < 3x \] This simplifies to: \[ 9 < 3x \]
Step 4: Divide by 3
\[ \frac{9}{3} < x \] This simplifies to: \[ 3 < x \] or expressed alternatively, \[ x > 3 \]
Step 5: Graph the solution
To graph the solution \( x > 3 \):
- Draw a number line.
- Put an open circle on the number 3 to indicate that 3 is not included in the solution set.
- Shade the line to the right of 3 to indicate that all values greater than 3 are included.
Thus, the solution to the inequality is \( x > 3 \), which you can represent graphically as described.