Question

learning goal from lesson 9.3: i can solve rational equations in one variable and determine extraneous solutions. i can solve radical equations in one variable and determine extraneous solutions. i can explain how extraneous solutions may arise from rational or radical equations. how i did (circle one) i got it! im still learning it. solve the rational equation algebraically. (1 point) 17. $\frac{2x}{x - 5}=\frac{3x^{2}-15x}{x^{2}-9x + 20}$ what is a possible first step to solving the rational expression? (1/2 points) 18. $\frac{7}{x - 4}=\frac{1}{2x}$ find the common denominators. cross multiply. use the quadratic formula. add the numerator and denominator. write a rational equation for the real - world application. then solve. (1 point) 19. a glassblower can produce several sets of simple glasses in about 3 hours. when the glassblower works with an apprentice, the job takes about 2 hours. write an equation that can be used to find the amount of time t, in hours, it would take for the apprentice to make the same number of sets of glasses when working alone.

Explanation:

Step1: Factor the denominators

The denominator of the right - hand side $x^{2}-9x + 20=(x - 4)(x - 5)$. The given rational equation is $\frac{2x}{x - 5}=\frac{3x^{2}-15x}{x^{2}-9x + 20}=\frac{3x(x - 5)}{(x - 4)(x - 5)}$.

Step2: Note the restrictions

The restrictions on the variable are $x
eq4$ and $x
eq5$ (to avoid division by zero).

Step3: Cross - multiply

Cross - multiplying gives $2x(x - 4)(x - 5)=3x(x - 5)(x - 5)$. Since $x
eq5$, we can divide both sides by $(x - 5)$ (for non - extraneous solutions). We get $2x(x - 4)=3x(x - 5)$.

Step4: Expand and simplify

Expand: $2x^{2}-8x=3x^{2}-15x$. Rearrange to get $3x^{2}-15x-2x^{2}+8x = 0$, which simplifies to $x^{2}-7x=0$. Factor out $x$: $x(x - 7)=0$. So $x = 0$ or $x=7$.

for question 18:
Cross - multiplying is a common first step when solving a rational equation of the form $\frac{a}{b}=\frac{c}{d}$. It simplifies the equation by getting rid of the fractions.

for question 19:
The glassblower's rate of work is $\frac{1}{3}$ (sets of glasses per hour). When working with the apprentice, their combined rate is $\frac{1}{2}$ (sets of glasses per hour). Let the apprentice's rate be $\frac{1}{t}$ (where $t$ is the time it takes the apprentice to complete the job alone). The equation for combined work rates is $\frac{1}{3}+\frac{1}{t}=\frac{1}{2}$.

Step1: Solve the equation

Subtract $\frac{1}{3}$ from both sides: $\frac{1}{t}=\frac{1}{2}-\frac{1}{3}$. Find a common denominator: $\frac{1}{t}=\frac{3 - 2}{6}=\frac{1}{6}$. So $t = 6$.

Answer:

$x = 0$ or $x = 7$