Answer :
- Solve the first inequality $x+3 < -2$ to get $x < -5$.
- Solve the second inequality $x-7 > 0$ to get $x > 7$.
- Combine the solutions using 'or' to get $x < -5$ or $x > 7$.
- The final answer is $x<-5$ or $x>7$, which corresponds to option c. $\boxed{x<-5 \text{ or } x>7}$
### Explanation
1. Understanding the problem
We are given the compound inequality $x+3<-2$ or $x-7>0$. We need to solve each inequality separately and then combine the solutions.
2. Solving the first inequality
First, let's solve the inequality $x+3<-2$. To isolate $x$, we subtract 3 from both sides of the inequality:$$x+3-3<-2-3$$$$x<-5$$
3. Solving the second inequality
Next, let's solve the inequality $x-7>0$. To isolate $x$, we add 7 to both sides of the inequality:$$x-7+7>0+7$$$$x>7$$
4. Combining the solutions
Since the compound inequality is connected by 'or', we take the union of the two solutions. Therefore, the solution to the compound inequality is $x<-5$ or $x>7$.
5. Final Answer
Comparing our solution $x<-5$ or $x>7$ with the given options, we see that it matches option c.
### Examples
Compound inequalities are used in various real-world scenarios, such as determining the acceptable range for a product's temperature during storage or transportation. For example, a certain medicine needs to be stored at a temperature below -5 degrees Celsius or above 7 degrees Celsius to maintain its effectiveness. Solving compound inequalities helps ensure that the storage conditions meet these requirements.
- Solve the second inequality $x-7 > 0$ to get $x > 7$.
- Combine the solutions using 'or' to get $x < -5$ or $x > 7$.
- The final answer is $x<-5$ or $x>7$, which corresponds to option c. $\boxed{x<-5 \text{ or } x>7}$
### Explanation
1. Understanding the problem
We are given the compound inequality $x+3<-2$ or $x-7>0$. We need to solve each inequality separately and then combine the solutions.
2. Solving the first inequality
First, let's solve the inequality $x+3<-2$. To isolate $x$, we subtract 3 from both sides of the inequality:$$x+3-3<-2-3$$$$x<-5$$
3. Solving the second inequality
Next, let's solve the inequality $x-7>0$. To isolate $x$, we add 7 to both sides of the inequality:$$x-7+7>0+7$$$$x>7$$
4. Combining the solutions
Since the compound inequality is connected by 'or', we take the union of the two solutions. Therefore, the solution to the compound inequality is $x<-5$ or $x>7$.
5. Final Answer
Comparing our solution $x<-5$ or $x>7$ with the given options, we see that it matches option c.
### Examples
Compound inequalities are used in various real-world scenarios, such as determining the acceptable range for a product's temperature during storage or transportation. For example, a certain medicine needs to be stored at a temperature below -5 degrees Celsius or above 7 degrees Celsius to maintain its effectiveness. Solving compound inequalities helps ensure that the storage conditions meet these requirements.
