1. **State the problem:** Solve the equation $$\frac{1}{2}|x + 1| + 1 = |x|$$ using the graph. 2. **Understand the functions:** The left side is $$y = \frac{1}{2}|x + 1| + 1$$, a V-shaped graph with vertex at $$(-1,1)$$ and slope $$\frac{1}{2}$$ on each side. The right side is $$y = |x|$$, a V-shaped graph with vertex at $$(0,0)$$ and slope $$1$$ on each side. 3. **Find intersection points:** The solutions to the equation are the x-values where the two graphs intersect. 4. **Analyze intervals:** Because of absolute values, consider cases: - Case 1: $$x \geq 0$$ - Then $$|x| = x$$ and $$|x+1| = x+1$$. - Equation becomes $$\frac{1}{2}(x+1) + 1 = x$$. - Simplify: $$\frac{1}{2}x + \frac{1}{2} + 1 = x$$. - Combine constants: $$\frac{1}{2}x + \frac{3}{2} = x$$. - Subtract $$\frac{1}{2}x$$ from both sides: $$\cancel{\frac{1}{2}x} + \frac{3}{2} = \cancel{\frac{1}{2}x} + \frac{1}{2}x$$. - This gives $$\frac{3}{2} = \frac{1}{2}x$$. - Multiply both sides by 2: $$3 = x$$. - Check if $$x=3 \geq 0$$: yes, valid solution. - Case 2: $$-1 \leq x < 0$$ - Then $$|x| = -x$$ and $$|x+1| = x+1$$. - Equation: $$\frac{1}{2}(x+1) + 1 = -x$$. - Simplify left: $$\frac{1}{2}x + \frac{1}{2} + 1 = -x$$. - Combine constants: $$\frac{1}{2}x + \frac{3}{2} = -x$$. - Add $$x$$ to both sides: $$\frac{1}{2}x + x + \frac{3}{2} = 0$$. - Combine x terms: $$\frac{3}{2}x + \frac{3}{2} = 0$$. - Subtract $$\frac{3}{2}$$: $$\frac{3}{2}x = -\frac{3}{2}$$. - Divide both sides by $$\frac{3}{2}$$: $$x = -1$$. - Check if $$-1 \leq x < 0$$: yes, valid solution. - Case 3: $$x < -1$$ - Then $$|x| = -x$$ and $$|x+1| = -(x+1) = -x -1$$. - Equation: $$\frac{1}{2}(-x -1) + 1 = -x$$. - Simplify left: $$-\frac{1}{2}x - \frac{1}{2} + 1 = -x$$. - Combine constants: $$-\frac{1}{2}x + \frac{1}{2} = -x$$. - Add $$\frac{1}{2}x$$ to both sides: $$\frac{1}{2} = -x + \frac{1}{2}x = -\frac{1}{2}x$$. - Multiply both sides by -2: $$-1 = x$$. - Check if $$x < -1$$: no, $$x = -1$$ is not less than -1, so no solution here. 5. **Final solutions:** $$x = -1$$ and $$x = 3$$. These match the intersection points of the two graphs. **Answer:** $$x = -1, x = 3$$