Rational equations: how to solve them

1. Restriction: $x \neq 0$. Multiply both sides by $x$.
   $$3 = 6x$$
2. Divide by 6.
   $$x = \frac{1}{2}$$
3. Check: $\frac{3}{1/2} = 6$.
   $$3 \div \tfrac{1}{2} = 6 \checkmark$$

Cevap: $x = \frac{1}{2}$

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1. Restriction: $x \neq 0$. The LCD of $x$ and 2 is $2x$; multiply every term by it.
   $$6 + x = 4x$$
2. Subtract $x$ from both sides.
   $$6 = 3x$$
3. Divide by 3.
   $$x = 2$$
4. Check.
   $$\tfrac{3}{2} + \tfrac{1}{2} = 2 \checkmark$$

Cevap: $x = 2$

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1. Restriction: $x \neq 3$. Multiply both sides by $x - 3$.
   $$x + 1 = 3(x - 3) = 3x - 9$$
2. Subtract $x$ and add 9.
   $$10 = 2x$$
3. Divide by 2. Since 5 ≠ 3, the answer is allowed.
   $$x = 5$$
4. Check.
   $$\tfrac{5 + 1}{5 - 3} = \tfrac{6}{2} = 3 \checkmark$$

Cevap: $x = 5$

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1. Restriction: $x \neq 2$. Multiply every term — including the 1 — by $x - 2$.
   $$x + (x - 2) = 2$$
2. Simplify and solve.
   $$2x - 2 = 2 \Rightarrow x = 2$$
3. But $x = 2$ makes the denominators zero, so it is an extraneous solution and must be rejected.
   $$\text{no solution}$$

Cevap: No solution ($x = 2$ is extraneous)

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1. Restrictions: $x \neq 0$ and $x \neq -2$. Multiply every term by the LCD, $4x(x + 2)$.
   $$4(x + 2) + 4x = 3x(x + 2)$$
2. Expand and collect everything on one side.
   $$3x^2 - 2x - 8 = 0$$
3. Factor.
   $$(3x + 4)(x - 2) = 0$$
4. Both answers are allowed.
   $$x = 2 \text{ or } x = -\tfrac{4}{3}$$
5. Check $x = 2$: $\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$. Check $x = -\frac{4}{3}$: $-\frac{3}{4} + \frac{3}{2} = \frac{3}{4}$.
   $$\checkmark\ \checkmark$$

Cevap: $x = 2$ or $x = -\frac{4}{3}$

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1. Cross-multiply.
   $$12x = 4 \times 9 = 36$$
2. Divide by 12.
   $$x = 3$$
3. Check: $\frac{3}{4}$ and $\frac{9}{12}$ are equal fractions.
   $$\tfrac{9}{12} = \tfrac{3}{4} \checkmark$$

Cevap: $x = 3$

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1. Let $t$ be the time together. Per hour, A fills $\frac{1}{3}$ and B fills $\frac{1}{6}$; together they fill $\frac{1}{t}$.
   $$\frac{1}{3} + \frac{1}{6} = \frac{1}{t}$$
2. Multiply every term by the LCD, $6t$.
   $$2t + t = 6$$
3. Solve.
   $$3t = 6 \Rightarrow t = 2$$
4. Check: in 2 hours A fills $\frac{2}{3}$ and B fills $\frac{1}{3}$ — one full tank.
   $$\tfrac{2}{3} + \tfrac{1}{3} = 1 \checkmark$$

Cevap: 2 hours

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