Questions: Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. (x+4)/(x+7)<2

Solution Steps

First, we solve the inequality: \[ \frac{x+4}{x+7}<2 \]

Subtract 2 from both sides: \[ \frac{x+4}{x+7} - 2 < 0 \]

Combine the terms over a common denominator: \[ \frac{x+4 - 2(x+7)}{x+7} < 0 \]

Simplify the numerator: \[ \frac{x+4 - 2x - 14}{x+7} < 0 \] \[ \frac{-x - 10}{x+7} < 0 \]

The critical points are where the numerator and denominator are zero: \[ -x - 10 = 0 \quad \Rightarrow \quad x = -10 \] \[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]

We test the intervals around the critical points \(x = -10\) and \(x = -7\):

1. \(x < -10\)
2. \(-10 < x < -7\)
3. \(x > -7\)

1. For \(x < -10\), choose \(x = -11\): \[ \frac{-(-11) - 10}{-11 + 7} = \frac{11 - 10}{-4} = \frac{1}{-4} < 0 \]
2. For \(-10 < x < -7\), choose \(x = -9\): \[ \frac{-(-9) - 10}{-9 + 7} = \frac{9 - 10}{-2} = \frac{-1}{-2} > 0 \]
3. For \(x > -7\), choose \(x = -6\): \[ \frac{-(-6) - 10}{-6 + 7} = \frac{6 - 10}{1} = \frac{-4}{1} < 0 \]

Final Answer