Chapter 11.5, Problem 1P

a.

To determine

To solve the equation $\frac{1}{3}+\frac{3}{x}=\frac{2}{x}$ and check the solution.

Answer to Problem 1P

  $x=-3$

Explanation of Solution

Given:

Given equation

  $\frac{1}{3}+\frac{3}{x}=\frac{2}{x}$

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
1. Multiply both sides of equations by LCD(least common denominator) of all denominators.
2. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

In order to solve the equation $\frac{1}{3}+\frac{3}{x}=\frac{2}{x}$ , observe that the LCD of all the denominators is $3x$ . So multiply both sides by this LCD to get rid of the denominators and simplify further as shown below

  $\begin{array}{c}\frac{1}{3}+\frac{3}{x}=\frac{2}{x}\\ \frac{1}{3}\cdot 3x+\frac{3}{x}\cdot 3x=\frac{2}{x}\cdot 3x\\ \frac{3x}{3}+\frac{9x}{x}=\frac{6x}{x}\\ x+9=6\\ x+9-9=6-9\\ x=-3\end{array}$

Thus, the solution of the given equation is $x=-3$ .

Now to check the solution, substitute $x=-3$ in the given equation and check whether left hand side value is equal to the right hand side value or no. So, it gives

  $\begin{array}{c}\frac{1}{3}+\frac{3}{-3}=\frac{2}{-3}\\ \frac{1}{3}-\frac{3}{3}=-\frac{2}{3}\\ \frac{1-3}{3}=-\frac{2}{3}\\ -\frac{2}{3}=-\frac{2}{3}\text{True Statement}\end{array}$

Thus, the solution is correct.

b.

To determine

To solve the equation $\frac{4}{7x}+\frac{1}{3}=\frac{7}{3x}$ and check the solution.

Answer to Problem 1P

  $x=\frac{37}{7}$

Explanation of Solution

Given:

Given equation

  $\frac{4}{7x}+\frac{1}{3}=\frac{7}{3x}$

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
1. Multiply both sides of equations by LCD(least common denominator) of all denominators.
2. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

In order to solve the equation $\frac{4}{7x}+\frac{1}{3}=\frac{7}{3x}$ , observe that the LCD of all the denominators is $21x$ . So multiply both sides by this LCD to get rid of the denominators and simplify further as shown below

  $\begin{array}{c}\frac{4}{7x}+\frac{1}{3}=\frac{7}{3x}\\ \frac{4}{7x}\cdot 21x+\frac{1}{3}\cdot 21x=\frac{7}{3x}\cdot 21x\\ 12+7x=49\\ 12+7x-12=49-12\\ 7x=37\\ x=\frac{37}{7}\end{array}$

Thus, the solution of the given equation is $x=\frac{37}{7}$ .

Now to check the solution, substitute $x=\frac{37}{7}$ in the given equation and check whether left hand side value is equal to the right hand side value or no. So, it gives

  $\begin{array}{c}\frac{4}{7\left(\frac{37}{7}\right)}+\frac{1}{3}=\frac{7}{3\left(\frac{37}{7}\right)}\\ \frac{4}{37}+\frac{1}{3}=\frac{49}{111}\\ \frac{12+37}{111}=\frac{49}{111}\\ \frac{49}{111}=\frac{49}{111}\text{True Statement}\end{array}$

Thus, the solution is correct.