Question

Solve and graph this inequality: 5(3 - x) < -2x +6

Answer 1

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 \( x > 3 \):
1. Draw a number line.
2. Put an open circle on the number 3 to indicate that 3 is not included in the solution set.
3. 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.