1. Given the data points: $x-2$, $x+5$, $x+3$, $3x-4$, $2x+7$, the mean is given as 7. 2. The formula for the mean of $n$ data points $a_1, a_2, ..., a_n$ is: $$\text{Mean} = \frac{a_1 + a_2 + ... + a_n}{n}$$ 3. Substitute the data points and mean value: $$7 = \frac{(x-2) + (x+5) + (x+3) + (3x-4) + (2x+7)}{5}$$ 4. Simplify the numerator: $$(x-2) + (x+5) + (x+3) + (3x-4) + (2x+7) = x - 2 + x + 5 + x + 3 + 3x - 4 + 2x + 7$$ Combine like terms: $$x + x + x + 3x + 2x = 8x$$ $$-2 + 5 + 3 - 4 + 7 = 9$$ So the sum is: $$8x + 9$$ 5. Set up the equation: $$7 = \frac{8x + 9}{5}$$ Multiply both sides by 5: $$35 = 8x + 9$$ 6. Solve for $x$: $$8x = 35 - 9 = 26$$ $$x = \frac{26}{8} = \frac{13}{4} = 3.25$$ 7. To find the median, first substitute $x=3.25$ into the data points: $$(3.25 - 2) = 1.25$$ $$(3.25 + 5) = 8.25$$ $$(3.25 + 3) = 6.25$$ $$(3 \times 3.25 - 4) = 9.75 - 4 = 5.75$$ $$(2 \times 3.25 + 7) = 6.5 + 7 = 13.5$$ 8. Arrange the data in ascending order: $$1.25, 5.75, 6.25, 8.25, 13.5$$ The median is the middle value (3rd value): $$\text{Median} = 6.25$$ 9. The mode is the value that appears most frequently. Since all values are distinct, there is no mode. Final answers: $$x = 3.25$$ $$\text{Median} = 6.25$$ $$\text{Mode} = \text{None}$$