Question

solving rational equations by cross multiplying:

1. \\(\frac{3}{x + 2} = \frac{6}{x - 1}\\)
2. \\(\frac{5}{x - 4} = \frac{10}{x + 3}\\)
3. \\(\frac{2}{x + 7} = \frac{8}{x - 5}\\)
4. \\(\frac{9}{x - 2} = \frac{3}{x + 4}\\)
5. \\(\frac{7}{x + 1} = \frac{14}{x - 9}\\)
6. \\(\frac{4}{x - 6} = \frac{12}{x + 2}\\)
7. \\(\frac{6}{x + 8} = \frac{9}{x - 1}\\)
8. \\(\frac{10}{x - 3} = \frac{4}{x + 5}\\)
9. \\(\frac{15}{x + 6} = \frac{5}{x - 2}\\)
10. \\(\frac{8}{x - 7} = \frac{12}{x + 1}\\)
11. \\(\frac{5}{x + 10} = \frac{20}{x - 6}\\)
12. \\(\frac{3}{x + 2} = \frac{6}{x - 1}\\)
13. \\(\frac{5}{2 + 3x} = \frac{10}{4 - x}\\)
14. \\(\frac{7}{-2x + 6} = \frac{14}{x + 3}\\)
15. \\(\frac{3}{4x + 1} = \frac{12}{-3x + 2}\\)
16. \\(\frac{10}{-x + 6} = \frac{20}{2x + 1}\\)
17. \\(\frac{15}{-4x + 2} = \frac{5}{x + 3}\\)

Explanation:

1) $\frac{3}{x+2}=\frac{6}{x-1}$

Step1: Cross multiply to eliminate denominators

$3(x-1)=6(x+2)$

Step2: Expand both sides

$3x-3=6x+12$

Step3: Isolate $x$ terms

$3x-6x=12+3$

Step4: Simplify and solve for $x$

$-3x=15 \implies x=-5$

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2) $\frac{5}{x-4}=\frac{10}{x+3}$

Step1: Cross multiply

$5(x+3)=10(x-4)$

Step2: Expand both sides

$5x+15=10x-40$

Step3: Isolate $x$ terms

$5x-10x=-40-15$

Step4: Simplify and solve for $x$

$-5x=-55 \implies x=11$

---

3) $\frac{2}{x+7}=\frac{8}{x-5}$

Step1: Cross multiply

$2(x-5)=8(x+7)$

Step2: Expand both sides

$2x-10=8x+56$

Step3: Isolate $x$ terms

$2x-8x=56+10$

Step4: Simplify and solve for $x$

$-6x=66 \implies x=-11$

---

4) $\frac{9}{x-2}=\frac{3}{x+4}$

Step1: Cross multiply

$9(x+4)=3(x-2)$

Step2: Expand both sides

$9x+36=3x-6$

Step3: Isolate $x$ terms

$9x-3x=-6-36$

Step4: Simplify and solve for $x$

$6x=-42 \implies x=-7$

---

5) $\frac{7}{x+1}=\frac{14}{x-9}$

Step1: Cross multiply

$7(x-9)=14(x+1)$

Step2: Expand both sides

$7x-63=14x+14$

Step3: Isolate $x$ terms

$7x-14x=14+63$

Step4: Simplify and solve for $x$

$-7x=77 \implies x=-11$

---

6) $\frac{4}{x-6}=\frac{12}{x+2}$

Step1: Cross multiply

$4(x+2)=12(x-6)$

Step2: Expand both sides

$4x+8=12x-72$

Step3: Isolate $x$ terms

$4x-12x=-72-8$

Step4: Simplify and solve for $x$

$-8x=-80 \implies x=10$

---

7) $\frac{6}{x+8}=\frac{9}{x-1}$

Step1: Cross multiply

$6(x-1)=9(x+8)$

Step2: Expand both sides

$6x-6=9x+72$

Step3: Isolate $x$ terms

$6x-9x=72+6$

Step4: Simplify and solve for $x$

$-3x=78 \implies x=-26$

---

8) $\frac{10}{x-3}=\frac{4}{x+5}$

Step1: Cross multiply

$10(x+5)=4(x-3)$

Step2: Expand both sides

$10x+50=4x-12$

Step3: Isolate $x$ terms

$10x-4x=-12-50$

Step4: Simplify and solve for $x$

$6x=-62 \implies x=-\frac{31}{3}$

---

9) $\frac{15}{x+6}=\frac{5}{x-2}$

Step1: Cross multiply

$15(x-2)=5(x+6)$

Step2: Expand both sides

$15x-30=5x+30$

Step3: Isolate $x$ terms

$15x-5x=30+30$

Step4: Simplify and solve for $x$

$10x=60 \implies x=6$

---

10) $\frac{8}{x-7}=\frac{12}{x+1}$

Step1: Cross multiply

$8(x+1)=12(x-7)$

Step2: Expand both sides

$8x+8=12x-84$

Step3: Isolate $x$ terms

$8x-12x=-84-8$

Step4: Simplify and solve for $x$

$-4x=-92 \implies x=23$

---

11) $\frac{5}{x+10}=\frac{20}{x-6}$

Step1: Cross multiply

$5(x-6)=20(x+10)$

Step2: Expand both sides

$5x-30=20x+200$

Step3: Isolate $x$ terms

$5x-20x=200+30$

Step4: Simplify and solve for $x$

$-15x=230 \implies x=-\frac{46}{3}$

---

12) $\frac{3}{x+2}=\frac{6}{x-1}$

Step1: Cross multiply

$3(x-1)=6(x+2)$

Step2: Expand both sides

$3x-3=6x+12$

Step3: Isolate $x$ terms

$3x-6x=12+3$

Step4: Simplify and solve for $x$

$-3x=15 \implies x=-5$

---

13) $\frac{5}{2+3x}=\frac{10}{4-x}$

Step1: Cross multiply

$5(4-x)=10(2+3x)$

Step2: Expand both sides

$20-5x=20+30x$

Step3: Isolate $x$ terms

$-5x-30x=20-20$

Step4: Simplify and solve for $x$

$-35x=0 \implies x=0$

---

14) $\frac{7}{-2x+6}=\frac{14}{x+3}$

Step1: Cross multiply

$7(x+3)=14(-2x+6)$

Step2: Expand both sides

$7x+21=-28x+84$

Step3: Isolate $x$ terms

$7x+28x=84-21$

Step4: Simplify and solve for $x$

$35x=63 \implies x=\frac{9}{5}$

---

15) $\frac{3}{4x+1}=\frac{12}{-3x+2}$

Step1: Cross multiply

$3(-3x+2)=12(4x+1)$

Step2: Expand both sides

$-9x+6=48x+12$

Step3: Isolate $x$ terms

$-9x-48x=12-6$

Step4: Simplify and solve for $x$

$-57x=6 \implies x=-\frac{2}{19}$

---

16) $\frac{10}{-x+6}=\frac{20}{2x+1}$

Step1: Cross multiply

$10(2x+1)=20(-x+6)$

Step2: Expand both sides

$20x+10=-20x+120$

Step3: Isolate $x$ terms

$20x+20x=120-10$

Step4: Simplify and solve for $x$

$40x=110 \implies x=\frac{11}{4}$

---

17) $\frac{15}{-4x+2}=\frac{5}{x+3}$

Step1: Cross multiply

$15(x+3)=5(-4x+2)$

Step2: Expand both sides

$15x+45=-20x+10$

Step3: Isolate $x$ terms

$15x+20x=10-45$

Step4: Simplify and solve for $x$

$35x=-35 \implies x=-1$

Answer:

1. $x=-5$
2. $x=11$
3. $x=-11$
4. $x=-7$
5. $x=-11$
6. $x=10$
7. $x=-26$
8. $x=-\frac{31}{3}$
9. $x=6$
10. $x=23$
11. $x=-\frac{46}{3}$
12. $x=-5$
13. $x=0$
14. $x=\frac{9}{5}$
15. $x=-\frac{2}{19}$
16. $x=\frac{11}{4}$
17. $x=-1$