1. **State the problem:** We have an L-shaped compound figure with total area 92 cm². The total width is 10 cm, total height is 8 cm. The top horizontal segment is 4 cm, the vertical segment next to it is $x$ cm. We need to find $x$. 2. **Understand the shape:** The compound shape can be divided into two rectangles: - Rectangle A: width 4 cm, height $x$ cm - Rectangle B: width (10 - 4) = 6 cm, height (8 - x) cm 3. **Write the area formula:** Total area = Area of Rectangle A + Area of Rectangle B $$\text{Area} = 4 \times x + 6 \times (8 - x)$$ 4. **Set up the equation:** Given total area is 92 cm², $$4x + 6(8 - x) = 92$$ 5. **Expand and simplify:** $$4x + 48 - 6x = 92$$ 6. **Combine like terms:** $$\cancel{4x} - \cancel{6x} = -2x$$ So, $$-2x + 48 = 92$$ 7. **Isolate $x$:** $$-2x = 92 - 48$$ $$-2x = 44$$ 8. **Divide both sides by -2:** $$x = \frac{44}{\cancel{-2}} \times \cancel{-1} = -22$$ Since length cannot be negative, check the sign carefully: Actually, $$-2x = 44 \implies x = \frac{44}{-2} = -22$$ This is negative, which is impossible for length. Re-examine step 6: Step 6 correction: $$4x + 48 - 6x = 92$$ $$4x - 6x = -2x$$ So, $$-2x + 48 = 92$$ Subtract 48 from both sides: $$-2x = 92 - 48 = 44$$ Divide both sides by -2: $$x = \frac{44}{-2} = -22$$ Negative length is impossible, so the error is in the assumption of the rectangles. 9. **Reconsider the rectangles:** The vertical segment adjacent to the 4 cm top segment is $x$ cm, so the height of the left rectangle is $x$ cm, width 4 cm. The right rectangle has width 6 cm and height (8 - x) cm. Area total: $$4x + 6(8 - x) = 92$$ Simplify: $$4x + 48 - 6x = 92$$ $$-2x + 48 = 92$$ Subtract 48: $$-2x = 44$$ Divide: $$x = -22$$ Negative again. 10. **Check if $x$ is the height of the right rectangle instead:** If $x$ is the height of the right rectangle, then left rectangle height is $8 - x$. Area: $$4(8 - x) + 6x = 92$$ Expand: $$32 - 4x + 6x = 92$$ $$2x + 32 = 92$$ Subtract 32: $$2x = 60$$ Divide: $$x = 30$$ But $x=30$ cm is greater than total height 8 cm, impossible. 11. **Conclusion:** The only possible interpretation is that $x$ is the vertical segment on the left side, and the area equation is correct but the problem data may have an inconsistency. 12. **Final answer:** Based on the problem statement and calculations, the value of $x$ is $-22$ cm, which is not physically possible. Please verify the problem data. **If we assume the total area is 92 cm² and the shape dimensions as given, the value of $x$ is:** $$\boxed{x = -22}$$ This suggests a possible error in the problem setup or measurements.