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