Question

Solve graphically by the intersection method. Give the solution in interval notation.
5 x minus 3 less than 2 x minus 6
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Part 1
Graph y 1 equals 5 x minus 3 and y 2 equals 2 x minus 6 using the graphing tool.

Part 2
What is the​ solution?

Answer 1

To solve the inequality \( 5x - 3 < 2x - 6 \) graphically by using the intersection method, we'll first rearrange the inequality into two equations:

1. \( y_1 = 5x - 3 \)
2. \( y_2 = 2x - 6 \)

Next, we should graph these two equations. Here is how you can do it:

### Step 1: Identify Points to Graph
- For \( y_1 = 5x - 3 \):
- When \( x = 0 \): \( y = -3 \) (Point: (0, -3))
- When \( x = 1 \): \( y = 2 \) (Point: (1, 2))
- When \( x = 2 \): \( y = 7 \) (Point: (2, 7))

- For \( y_2 = 2x - 6 \):
- When \( x = 0 \): \( y = -6 \) (Point: (0, -6))
- When \( x = 3 \): \( y = 0 \) (Point: (3, 0))
- When \( x = 5 \): \( y = 4 \) (Point: (5, 4))

### Step 2: Graph the Lines
Using the points above, you can plot the lines on a graph. You would draw a straight line for each equation, \( y_1 \) and \( y_2 \).

### Step 3: Find the Intersection
To find the intersection of the two lines \( y_1 = 5x - 3 \) and \( y_2 = 2x - 6 \), set them equal to each other:

\[
5x - 3 = 2x - 6
\]

### Step 4: Solve for \( x \)
Subtract \( 2x \) from both sides:

\[
5x - 2x - 3 = -6
\]
\[
3x - 3 = -6
\]
Add 3 to both sides:

\[
3x = -3
\]
Divide by 3:

\[
x = -1
\]

### Step 5: Find the \( y \) value at the Intersection
Now substitute \( x = -1 \) back into either equation to find \( y \):

\[
y = 5(-1) - 3 = -5 - 3 = -8
\]

So the intersection point is \((-1, -8)\).

### Step 6: Determine the Range of \( x \) for the Inequality
Since we're looking for \( 5x - 3 < 2x - 6 \), we determine:

- For \( x < -1 \): \( y_1 \) will be below \( y_2 \).
- For \( x = -1 \): \( y_1 = y_2 \).
- For \( x > -1 \): \( y_1 \) will be above \( y_2 \).

Thus, the solution to the inequality \( 5x - 3 < 2x - 6 \) occurs where \( x < -1 \).

### Final Solution in Interval Notation
The solution is:

\[
(-\infty, -1)
\]