intercepts of f(x)=(2x+9)/(3x-2)

On the horizontal axis, y = 0: solve

\frac{2 x + 9}{3 x - 2} = 0

1) Let's solve in

ℝ

\frac{2 x + 9}{3 x - 2} = 0

A quotient is zero if and only if its numerator is zero and its denominator is not: we solve the numerator equal to 0

⇥1.1) Let's solve in

ℝ

2 x + 9 = 0

⇥

2 x = -9

⇥ⓘ Both sides are divided by

2

⇥Candidates, pending the domain check:

x = - \frac{9}{2}

⇥1.2) To prevent the denominator from being zero, let's solve:

3 x - 2 \neq 0

⇥

3 x \neq 2

⇥ⓘ Both sides are divided by

3

⇥Domain condition:

x \neq \frac{2}{3}

For the expression

\frac{2 x + 9}{3 x - 2}

to exist, you must check:

3 x - 2 \neq 0

Domain constraints:

x \neq \frac{2}{3}

The graph crosses the horizontal axis at

(- \frac{9}{2}, 0)

On the vertical axis, x = 0: calculate f(0)

⇥

\frac{2 \cdot 0 + 9}{3 \cdot 0 - 2}

⇥ⓘ A product with a zero factor is zero:

2 \cdot 0

0

3 \cdot 0

0

⇥

\frac{0 + 9}{0 - 2}

⇥ⓘ Adding 0 changes nothing:

0 + 9

9

0 - 2

-2

■The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{9}{-2}

- \frac{9}{2}

⇥

- \frac{9}{2}

▶Axis intercepts:

{(- \frac{9}{2}, 0), (0, - \frac{9}{2})}