Questions: Solve the compound inequality. 4x + 6 < -6 or 3x + 5 > 14

Solve the compound inequality. 4x + 6 < -6 or 3x + 5 > 14
Transcript text: Solve the compound inequality. \[ 4 x+6<-6 \text { or } 3 x+5>14 \]
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Solution

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Solution Steps

To solve the compound inequality, we need to solve each inequality separately. For the first inequality, \(4x + 6 < -6\), we will isolate \(x\) by subtracting 6 from both sides and then dividing by 4. For the second inequality, \(3x + 5 > 14\), we will isolate \(x\) by subtracting 5 from both sides and then dividing by 3. The solution to the compound inequality will be the union of the solutions to the individual inequalities.

Step 1: Solve the First Inequality

We start with the first inequality \(4x + 6 < -6\). By isolating \(x\), we subtract 6 from both sides to get \(4x < -12\). Dividing both sides by 4 gives us \(x < -3\). Thus, the solution for the first inequality is \( (-\infty, -3) \).

Step 2: Solve the Second Inequality

Next, we solve the second inequality \(3x + 5 > 14\). We isolate \(x\) by subtracting 5 from both sides, resulting in \(3x > 9\). Dividing both sides by 3 yields \(x > 3\). Therefore, the solution for the second inequality is \( (3, \infty) \).

Final Answer

The solutions to the compound inequality are:

  1. From the first inequality: \( (-\infty, -3) \)
  2. From the second inequality: \( (3, \infty) \)

Thus, the final answer is: \[ \boxed{(-\infty, -3) \cup (3, \infty)} \]

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