Question

Solve graphically by the intersection method. Give the solution in interval notation.
6 x minus 3 less than 2 x minus 7
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Part 1
Graph y 1 equals 6 x minus 3 and y 2 equals 2 x minus 7 using the graphing tool.

Part 2
What is the​ solution?
  
enter your response here
​(Type your answer in interval​ notation.)

Answer 1

To solve the inequality \( 6x - 3 < 2x - 7 \) graphically, we will first rewrite it in a form that allows us to express it as a comparison of two functions. Specifically, we can rewrite the inequality as:

\[
y_1 = 6x - 3
\]
\[
y_2 = 2x - 7
\]

### Part 1: Graphing

You will need to graph the two equations:
1. **For \( y_1 = 6x - 3 \)**:
- The y-intercept is \(-3\).
- The slope is \(6\), which means for every unit you move to the right (increase x by 1), you move up 6 units.

2. **For \( y_2 = 2x - 7 \)**:
- The y-intercept is \(-7\).
- The slope is \(2\), meaning for every unit you increase x by 1, you move up 2 units.

### Finding the intersection

To find the intersection point of the two lines \( y_1 \) and \( y_2 \), set them equal to each other:

\[
6x - 3 = 2x - 7
\]

Now, solve for \( x \):
\[
6x - 2x = -7 + 3
\]
\[
4x = -4
\]
\[
x = -1
\]

Now substitute \( x = -1 \) into either equation to find \( y \):
Using \( y_1 \):
\[
y_1 = 6(-1) - 3 = -6 - 3 = -9
\]

Thus, the intersection point is \( (-1, -9) \).

### Part 2: Intervals

Next, we need to determine where \( 6x - 3 < 2x - 7 \) or equivalently where \( y_1 < y_2 \) on the graph.

1. Choose test points to the left, directly at, and to the right of \( x = -1 \).
- **Test Point \( x = -2 \)**:
- \( y_1 = 6(-2) - 3 = -12 - 3 = -15 \)
- \( y_2 = 2(-2) - 7 = -4 - 7 = -11 \)
- Here, \( -15 < -11 \) (True).

- **Test Point \( x = 0 \)**:
- \( y_1 = 6(0) - 3 = -3 \)
- \( y_2 = 2(0) - 7 = -7 \)
- Here, \( -3 < -7 \) (False).

2. Since the inequality holds true for \( x < -1 \) and false for \( x > -1 \), we can conclude that the solution to the inequality \( 6x - 3 < 2x - 7 \) is:

### Solution in interval notation:

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

Thus, the answer is:

\[
\boxed{(-\infty, -1)}
\]