d)) Use algebra tiles to find \( \left(-9 x^{2}+9 x\right)-\left(-9 x^{2}+x\right) \)

To solve \( \left(-9x^{2} + 9x\right) - \left(-9x^{2} + x\right) \) using **algebra tiles**, let's break down the problem step-by-step. ### Understanding Algebra Tiles Algebra tiles are visual representations used to model algebraic expressions. Each term in the expression is represented by a specific tile: - **Positive \(x^{2}\) tile**: Typically a large square. - **Negative \(x^{2}\) tile**: A large square of a different color (e.g., red). - **Positive \(x\) tile**: A rectangle. - **Negative \(x\) tile**: A rectangle of a different color (e.g., red). - **Positive constant tile**: A small square. - **Negative constant tile**: A small square of a different color (e.g., red). ### Step-by-Step Solution 1. **Represent Each Expression with Tiles:** - **First Expression:** \( -9x^{2} + 9x \) - **-9 \(x^{2}\) Tiles:** 9 large red squares. - **+9 \(x\) Tiles:** 9 rectangles. - **Second Expression:** \( -9x^{2} + x \) - **-9 \(x^{2}\) Tiles:** 9 large red squares. - **+1 \(x\) Tile:** 1 rectangle. 2. **Set Up the Subtraction:** \[ \left(-9x^{2} + 9x\right) - \left(-9x^{2} + x\right) \] Subtracting the second expression is equivalent to adding its additive inverse: \[ \left(-9x^{2} + 9x\right) + \left(+9x^{2} - x\right) \] 3. **Combine Like Terms:** - **\(x^{2}\) Terms:** - \( -9x^{2} + 9x^{2} = 0x^{2} \) - The large red squares cancel each other out. - **\(x\) Terms:** - \( +9x - x = +8x \) - You are left with 8 positive \(x\) rectangles. 4. **Final Representation:** After cancellation, the only tiles remaining are **8 positive \(x\)** rectangles. ### Conclusion Using algebra tiles, subtracting \( \left(-9x^{2} + x\right) \) from \( \left(-9x^{2} + 9x\right) \) results in: \[ 8x \] **Answer:** \(8x\)