Question

teacher: thursday solve the equation: $34 = 2m - 8$ expand the following: $\frac{2}{9}(\frac{9}{4}x - 9)$ simplifying the following expression: $6x - \frac{2}{3}(6x - 15y) + (-10)$ solve: $6m + 5 = -(1 - 8m)$ which is cheaper for a $65 item that costs $7.25 to ship? a 10% coupon off the cost of the item or free shipping? solve: $7 - 8x > -41$ write an inequality that represents the graph below: graph omitted solve and graph the inequality on a number line: $-2w geq -10$

Explanation:

Step1: Solve $34=2m-8$

Add 8 to both sides: $34+8=2m$ → $42=2m$. Divide by 2: $m=\frac{42}{2}=21$

Step2: Expand $\frac{2}{9}(\frac{9}{4}x - 9)$

Distribute: $\frac{2}{9} \cdot \frac{9}{4}x - \frac{2}{9} \cdot 9$ → $\frac{1}{2}x - 2$

Step3: Simplify $6x-\frac{2}{3}(6x-15y)+(-10)$

Distribute: $6x - 4x + 10y -10$. Combine like terms: $2x+10y-10$

Step4: Solve $6m+5=-(1-8m)$

Expand right side: $6m+5=-1+8m$. Subtract $6m$: $5=-1+2m$. Add 1: $6=2m$ → $m=3$

Step5: Compare costs

10% coupon: $65 \cdot 0.9 + 7.25 = 58.5 +7.25=65.75$. Free shipping: $65$. $65<65.75$, so free shipping is cheaper.

Step6: Solve $7-8x > -41$

Subtract 7: $-8x > -48$. Divide by -8 (reverse inequality): $x < 6$

Step7: Write inequality from graph

Closed circle at -1, arrow left: $x \leq -1$

Step8: Solve $-2w \geq -10$

Divide by -2 (reverse inequality): $w \leq 5$. Graph: closed circle at 5, arrow pointing left.

Answer:

1. $m=21$
2. $\frac{1}{2}x - 2$
3. $2x + 10y - 10$
4. $m=3$
5. Free shipping is cheaper.
6. $x < 6$
7. $x \leq -1$
8. $w \leq 5$ (graph: closed circle at 5, arrow left)