Precalculus Examples

$\frac{x + 7}{x - 19} \geq 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{x + 7}{x - 19} - 1 \geq 0$

Step 2

Simplify $\frac{x + 7}{x - 19} - 1$ .

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Step 2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 19}{x - 19}$ .

$\frac{x + 7}{x - 19} - 1 \cdot \frac{x - 19}{x - 19} \geq 0$

Step 2.2

Combine $- 1$ and $\frac{x - 19}{x - 19}$ .

$\frac{x + 7}{x - 19} + \frac{- (x - 19)}{x - 19} \geq 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x + 7 - (x - 19)}{x - 19} \geq 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{x + 7 - x - - 19}{x - 19} \geq 0$

Step 2.4.2

Multiply $- 1$ by $- 19$ .

$\frac{x + 7 - x + 19}{x - 19} \geq 0$

Step 2.4.3

Subtract $x$ from $x$ .

$\frac{0 + 7 + 19}{x - 19} \geq 0$

Step 2.4.4

Add $0$ and $7$ .

$\frac{7 + 19}{x - 19} \geq 0$

Step 2.4.5

Add $7$ and $19$ .

$\frac{26}{x - 19} \geq 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x - 19 = 0$

Step 4

Add $19$ to both sides of the equation.

$x = 19$

Step 5

Find the domain of $\frac{26}{x - 19}$ .

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Step 5.1

Set the denominator in $\frac{26}{x - 19}$ equal to $0$ to find where the expression is undefined.

$x - 19 = 0$

Step 5.2

Add $19$ to both sides of the equation.

$x = 19$

Step 5.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, 19) \cup (19, \infty)$

Step 6

The solution consists of all of the true intervals.

$x > 19$

Step 7

Convert the inequality to interval notation.

$(19, \infty)$

Step 8