Algebra › Rational equations

Equations with reciprocals

Solve the first problem 17 problems

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1. Finding a parameter from a known root ●○○○○ 10 ptsIf the solution of the equation $\frac{3}{10}a+\frac{2x+4}{2}=4(x-1)$ is $x=3$, find the value of $a$.
2. Unknown parameter from a root ●○○○○ 10 ptsIf $x=3$ is a root of the equation $\frac{a-2}{x}-\frac{1}{x-2}=0$, find the value of $a$.
3. Solve a quadratic with fractional coefficient ●○○○○ 10 ptsSolve $\dfrac{2}{3}x^2 - 4 = 12$ for $x$. Give both solutions, rounded to two decimal places.
4. Nested rational equation with a single root ●●○○○ 20 ptsSolve the equation $x+\frac{x}{x}+\frac{x}{x+\frac{x}{x}}=1$ and give the value of $x$.
5. Equation with reciprocals of linear factors ●●○○○ 20 ptsFind the roots of $\frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30}$, with $x \neq -4$ and $x \neq 7$.
6. Difference of fractions equals one ●●○○○ 20 ptsFind all values of $x$ (with $x \neq 0, 1, -1$) for which $\frac{1}{x^2 - x} - \frac{1}{x^3 - x} = 1$. Round each value to two decimal places.
7. Parameter values with no solution ●●●○○ 50 ptsDetermine all real numbers $a$ for which the equation $\frac{a}{1-\frac{1}{1+\frac{1}{x}}}=2023 x$ has no solutions. Enter the values in increasing order.

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