Practice 2 Solutions

Simplify. State when the given expression is undefined.

$\frac{3 x}{x^{2} + 7 x + 10} \cdot \frac{x + 2}{x^{2} + 5 x}$

$\frac{3 x}{(x + 5) (x + 2)} \cdot \frac{(x + 2)}{x (x + 5)} \frac{3 x}{(x + 5) (x + 2)} \cdot \frac{(x + 2)}{x (x + 5)}$

$\frac{3}{{(x + 5)}^{2}}, x \neq - 5, - 2, 0$

$\frac{b^{2} - 9 b + 14}{3 b + 18} \div \frac{3 b^{2} - 15 b - 72}{6 b + 36} \cdot \frac{b^{2} + 6 b + 9}{b^{2} - b - 2}$

$\frac{(b - 7) (b - 2)}{3 (b + 6)} \div \frac{3 (b^{2} - 5 b - 24)}{6 (b + 6)} \cdot \frac{(b + 3) (b + 3)}{(b - 2) (b + 1)} \frac{(b - 7) (b - 2)}{3 (b + 6)} \cdot \frac{6 (b + 6)}{3 (b - 8) (b + 3)} \cdot \frac{(b + 3) (b + 3)}{(b - 2) (b + 1)} \frac{(b - 7) (b - 2)}{3 (b + 6)} \cdot \frac{{}^{2}6 (b + 6)}{3 (b - 8) (b + 3)} \cdot \frac{(b + 3) (b + 3)}{(b - 2) (b + 1)}$

$\frac{2 (b - 7) (b + 3)}{3 (b - 8) (b + 1)}, b \neq - 6, - 3, - 1, 2, 8$

$\frac{2 n - 3}{n^{3} + 4 n^{2} + n + 4} \div \frac{4 n^{2} - 9}{4 n^{2} + 22 n + 24}$

$\frac{(2 n - 3)}{n^{2} (n + 4) + 1 (n + 4)} \div \frac{(2 n + 3) (2 n - 3)}{2 (2 n^{2} + 11 n + 12)} \frac{(2 n - 3)}{(n^{2} + 1) (n + 4)} \cdot \frac{2 (2 n + 3) (n + 4)}{(2 n + 3) (2 n - 3)} \frac{(2 n - 3)}{(n^{2} + 1) (n + 4)} \cdot \frac{2 (2 n + 3) (n + 4)}{(2 n + 3) (2 n - 3)}$

$\frac{2}{n^{2} + 1}, n \neq - 4, \pm \frac{3}{2}$

Note

For the expression $n^{2} + 1$ there are no restrictions in the set of real numbers. You will learn more about the square root of negative numbers in later lessons.

$\frac{k^{2} - j^{2}}{k - j} \div \frac{k^{4} - j^{4}}{k^{3} - j^{3}}$

$\frac{(k + j) (k - j)}{(k - j)} \div \frac{(k^{2} + j^{2}) (k^{2} - j^{2})}{(k - j) (k^{2} + k j + j^{2})} \frac{(k + j) (k - j)}{(k - j)} \cdot \frac{(k - j) (k^{2} + k j + j^{2})}{(k^{2} + j^{2}) (k + j) (k - j)} \frac{(k + j) (k - j)}{(k - j)} \cdot \frac{(k - j) (k^{2} + k j + j^{2})}{(k^{2} + j^{2}) (k + j) (k - j)}$

$\frac{k^{2} + k j + j^{2}}{k^{2} + j^{2}}, k \neq \pm j$

Note

$k^{2} + j^{2}$ cannot equal zero in the real number system.

Simplify.

$\frac{\frac{x^{2} + 4 x - 5}{x^{2} - 3 x - 18}}{\frac{x^{2} + 6 x + 5}{x^{2} - 4 x - 12}}$

$\frac{x^{2} + 4 x - 5}{x^{2} - 3 x - 18} \div \frac{x^{2} + 6 x + 5}{x^{2} - 4 x - 12} \frac{(x + 5) (x - 1)}{(x - 6) (x + 3)} \div \frac{(x + 5) (x + 1)}{(x - 6) (x + 2)} \frac{(x + 5) (x - 1)}{(x - 6) (x + 3)} \cdot \frac{(x - 6) (x + 2)}{(x + 5) (x + 1)} \frac{(x + 5) (x - 1)}{(x - 6) (x + 3)} \cdot \frac{(x - 6) (x + 2)}{(x + 5) (x + 1)} x \neq - 5, - 3, - 2, - 1, 6$

$\frac{(x - 1) (x + 2)}{(x + 3) (x + 1)}, x \neq - 5, - 3, - 2, - 1, 6$

$\frac{3 x^{2} + x - 14}{2 x^{2} + 13 x + 20} \cdot \frac{3 x^{2} - 48}{10 x^{2} - 24 x + 8} \div \frac{18 x^{2} - 98}{10 x^{2} + 21 x - 10}$

$\frac{(3 x + 7) (x - 2)}{(2 x + 5) (x + 4)} \cdot \frac{3 (x^{2} - 16)}{2 (5 x^{2} - 12 x + 4)} \div \frac{2 (9 x^{2} - 49)}{(2 x + 5) (5 x - 2)} \frac{(3 x + 7) (x - 2)}{(2 x + 5) (x + 4)} \cdot \frac{3 (x + 4) (x - 4)}{2 (5 x - 2) (x - 2)} \div \frac{2 (3 x + 7) (3 x - 7)}{(2 x + 5) (5 x - 2)} \frac{(3 x + 7) (x - 2)}{(2 x + 5) (x + 4)} \cdot \frac{3 (x + 4) (x - 4)}{2 (5 x - 2) (x - 2)} \cdot \frac{(2 x + 5) (5 x - 2)}{2 (3 x + 7) (3 x - 7)}$

$\frac{(3 x + 7) (x - 2)}{(2 x + 5) (x + 4)} \cdot \frac{3 (x + 4) (x - 4)}{2 (5 x - 2) (x - 2)} \cdot \frac{(2 x + 5) (5 x - 2)}{2 (3 x + 7) (3 x - 7)} x \neq - 4, - \frac{5}{2}, \pm \frac{7}{3}, \frac{2}{5}, 2$

$\frac{3 (x - 4)}{4 (3 x - 7)}, x \neq - 4, - \frac{5}{2}, \pm \frac{7}{3}, \frac{2}{5}, 2$

$\frac{8 h^{3} - 1}{6 h^{2} + 6 h} \cdot \frac{6 h^{4} + 12 h^{3} + 6 h^{2}}{4 h^{2} - 1}$

$\frac{(2 h - 1) (4 h^{2} + 2 h + 1)}{6 h (h + 1)} \cdot \frac{6 h^{2} (h^{2} + 2 h + 1)}{(2 h + 1) (2 h - 1)} \frac{(2 h - 1) (4 h^{2} + 2 h + 1)}{6 h (h + 1)} \cdot \frac{6 h^{2} (h + 1) (h + 1)}{(2 h + 1) (2 h - 1)} \frac{(2 h - 1) (4 h^{2} + 2 h + 1)}{6 h (h + 1)} \cdot \frac{6 h^{2} (h + 1) (h + 1)}{(2 h + 1) (2 h - 1)} h \neq - 1, 0, \pm \frac{1}{2}$

$\frac{h (h + 1) (4 h^{2} + 2 h + 1)}{(2 h + 1)}, h \neq - 1, 0, \pm \frac{1}{2}$

$\frac{r^{2} + 3 r - 10}{r^{2} + 6 r + 9} \div \frac{r^{2} + r - 20}{20 r^{2} - 11 r - 3} \div \frac{5 r^{2} - 9 r - 2}{r^{2} - r - 12}$

$\frac{(r + 5) (r - 2)}{(r + 3) (r + 3)} \div \frac{(r + 5) (r - 4)}{(4 r - 3) (5 r + 1)} \div \frac{(5 r + 1) (r - 2)}{(r + 3) (r - 4)} \frac{(r + 5) (r - 2)}{(r + 3) (r + 3)} \cdot \frac{(4 r - 3) (5 r + 1)}{(r + 5) (r - 4)} \cdot \frac{(r + 3) (r - 4)}{(5 r + 1) (r - 2)} \frac{(r + 5) (r - 2)}{(r + 3) (r + 3)} \cdot \frac{(4 r - 3) (5 r + 1)}{(r + 5) (r - 4)} \cdot \frac{(r + 3) (r - 4)}{(5 r + 1) (r - 2)} r \neq - 5, - 3, - \frac{1}{5}, \frac{3}{4}, 4$

$\frac{4 r - 3}{r + 3}, r \neq - 5, - 3, - \frac{1}{5}, \frac{3}{4}, 4$

$\frac{4 w^{2}}{w^{2} + 8 w + 16} \div \frac{16}{5 w^{2} - 80}$

$\frac{4 w^{2}}{(w + 4) (w + 4)} \div \frac{16}{5 (w^{2} - 16)} \frac{4 w^{2}}{(w + 4) (w + 4)} \div \frac{16}{5 (w + 4) (w - 4)} \frac{4 w^{2}}{(w + 4) (w + 4)} \cdot \frac{5 (w + 4) (w - 4)}{16} \frac{4 w^{2}}{(w + 4) (w + 4)} \cdot \frac{5 (w + 4) (w - 4)}{16_{4}} w \neq \pm 4$

$\frac{5 w^{2} (w - 4)}{4 (w + 4)}, w \neq \pm 4$

$\frac{\frac{y^{2} - 5 y + 6}{y^{2} + 3 y + 2}}{\frac{y^{2} - 2 y - 3}{y^{2} - 4}}$

$\frac{y^{2} - 5 y + 6}{y^{2} + 3 y + 2} \div \frac{y^{2} - 2 y - 3}{y^{2} - 4} \frac{(y - 2) (y - 3)}{(y + 1) (y + 2)} \div \frac{(y - 3) (y + 1)}{(y + 2) (y - 2)} \frac{(y - 2) (y - 3)}{(y + 1) (y + 2)} \cdot \frac{(y + 2) (y - 2)}{(y - 3) (y + 1)} \frac{(y - 2) (y - 3)}{(y + 1) (y + 2)} \cdot \frac{(y + 2) (y - 2)}{(y - 3) (y + 1)} y \neq \pm 2, - 1, 3$

$\frac{{(y - 2)}^{2}}{{(y - 1)}^{2}}, y \neq \pm 2, - 1, 3$

Find the length of a rectangle when the width is $\frac{x^{2} + 2 x + 1}{x^{2} + 8 x + 15}$ meters and the area is $\frac{x^{2} - 1}{x^{2} + 10 x + 25} {meters}^{2}$ . Explain whether x = 1 is a possible solution.

$A = l w A \div w = l l = \frac{x^{2} - 1}{x^{2} + 10 x + 25} \div \frac{x^{2} + 2 x + 1}{x^{2} + 8 x + 15} l = \frac{(x + 1) (x - 1)}{(x + 5) (x + 5)} \div \frac{(x + 1) (x + 1)}{(x + 5) (x + 3)} l = \frac{(x + 1) (x - 1)}{(x + 5) (x + 5)} \cdot \frac{(x + 5) (x + 3)}{(x + 1) (x + 1)} l = \frac{(x + 1) (x - 1)}{(x + 5) (x + 5)} \cdot \frac{(x + 5) (x + 3)}{(x + 1) (x + 1)} x \neq - 5, - 3, - 1$

$l = \frac{(x - 1) (x + 3)}{(x + 5) (x + 1)} meters$

Yes, x = 1 because it is not a restriction (or excluded value).

Find the area of a triangle with base $\frac{2 x^{2} + 5 x + 3}{x - 3}$ feet and the height of $\frac{x^{2} - x - 6}{x^{2} - 1}$ feet. Then determine if a length of x = 3 is possible. Explain your reasoning.

$A = \frac{1}{2} b h A = \frac{1}{2} \cdot \frac{2 x^{2} + 5 x + 3}{x - 3} \cdot \frac{x^{2} - x - 6}{x^{2} - 1} A = \frac{1}{2} \cdot \frac{(2 x + 3) (x + 1)}{(x - 3)} \cdot \frac{(x - 3) (x + 2)}{(x + 1) (x - 1)} A = \frac{1}{2} \cdot \frac{(2 x + 3) (x + 1)}{(x - 3)} \cdot \frac{(x - 3) (x + 2)}{(x + 1) (x - 1)} x \neq \pm 1, 3$

$A = \frac{(2 x + 3) (x + 2)}{2 (x - 1)} {ft}^{2}$

It is not possible for x = 3 because it would cause the denominator of the area to equal zero.

When rational expressions are divided, the reciprocal of the term after the division symbol must be written.

When a rational expression is undefined , the denominator has a value of zero.