The image shows a graph of the absolute value function $y = |3 - 2x|$, and you are asked to solve the equation $|3 - 2x| = 7 + x$.

Steps to solve $|3 - 2x| = 7 + x$:

1. Handle the absolute value function: The equation involves an absolute value, so we need to consider two cases:

   • Case 1: $3 - 2x \geq 0$, which gives $|3 - 2x| = 3 - 2x$.
   • Case 2: $3 - 2x < 0$, which gives $|3 - 2x| = -(3 - 2x) = -3 + 2x$.

Case 1: $3 - 2x \geq 0$ (i.e., $x \leq \frac{3}{2}$)

The equation becomes: $3 - 2x = 7 + x$ Solve for $x$: $3 - 2x = 7 + x$ Subtract $x$ from both sides: $3 - 3x = 7$ Subtract 3 from both sides: $-3x = 4$ Divide by $-3$: $x = -\frac{4}{3}$ Since $-\frac{4}{3} \leq \frac{3}{2}$, this solution is valid for this case.

Case 2: $3 - 2x < 0$ (i.e., $x > \frac{3}{2}$)

The equation becomes: $-3 + 2x = 7 + x$ Solve for $x$: $-3 + 2x = 7 + x$ Subtract $x$ from both sides: $-3 + x = 7$ Add 3 to both sides: $x = 10$ Since $10 > \frac{3}{2}$, this solution is valid for this case.

Final Solution:

The solutions are: $x = -\frac{4}{3} \quad \text{or} \quad x = 10$

Would you like further details on the steps, or have any additional questions?

Related Questions:

1. How do you graph $y = |3 - 2x|$ from the equation?
2. Can you solve a similar equation like $|x - 4| = 2x + 1$?
3. How do you handle absolute value inequalities like $|3 - 2x| \geq 7$?
4. How does the slope of the lines change when we break the absolute value equation?
5. How would the solution change if the right-hand side of the equation were $7 - x$ instead of $7 + x$?

Tip:

When solving absolute value equations, always remember to check if the solutions fall within the correct range for each case to avoid extraneous solutions.