Solve the equations graphically by sketching the graphs of the corresponding functions |4𝑥 + 2| = 3𝑥 + 5
To solve the equation |4𝑥 + 2| = 3𝑥 + 5 graphically, we need to sketch the graphs of the corresponding functions and find their point of intersection.
First, let's graph the function y = |4𝑥 + 2| by considering two cases: when 4𝑥 + 2 is positive and when it is negative.
Case 1: 4𝑥 + 2 > 0
In this case, the equation |4𝑥 + 2| = 4𝑥 + 2 holds true.
To graph this function, we can consider the function y = 4𝑥 + 2.
When 4𝑥 + 2 > 0, we have:
4𝑥 + 2 > 0
4𝑥 > -2
𝑥 > -1/2
So for values of 𝑥 greater than -1/2, we can use the function y = 4𝑥 + 2.
Case 2: 4𝑥 + 2 < 0
In this case, the equation |4𝑥 + 2| = -(4𝑥 + 2) holds true.
To graph this function, we can consider the function y = -(4𝑥 + 2).
When 4𝑥 + 2 < 0, we have:
4𝑥 + 2 < 0
4𝑥 < -2
𝑥 < -1/2
So for values of 𝑥 less than -1/2, we can use the function y = -(4𝑥 + 2).
Now, let's graph these functions on the same coordinate system:
For 𝑥 > -1/2:
- Choose some values of 𝑥 greater than -1/2 (for example, 𝑥 = 0, 1, 2).
- Plug these values into the equation 𝑦 = 4𝑥 + 2 to find the corresponding 𝑦-values.
- Plot these points on the graph.
For 𝑥 < -1/2:
- Choose some values of 𝑥 less than -1/2 (for example, 𝑥 = -2, -3, -4).
- Plug these values into the equation 𝑦 = -(4𝑥 + 2) to find the corresponding 𝑦-values.
- Plot these points on the graph.
Now, let's graph the function y = 3𝑥 + 5:
- Choose some values of 𝑥 (for example, 𝑥 = 0, 1, 2).
- Plug these values into the equation 𝑦 = 3𝑥 + 5 to find the corresponding 𝑦-values.
- Plot these points on the graph.
The graph of y = |4𝑥 + 2| (when 4𝑥 + 2 > 0) will be a line with a positive slope starting from the point (-1/2, 1) and extending to the right.
The graph of y = |4𝑥 + 2| (when 4𝑥 + 2 < 0) will be a line with a negative slope starting from the point (-1/2, -1) and extending to the left.
The graph of y = 3𝑥 + 5 will be a line with a positive slope that intersects the graph of y = |4𝑥 + 2|.
The point of intersection between the two graphs represents the solution to the equation |4𝑥 + 2| = 3𝑥 + 5.