Solve the compound inequality.
[tex]7 x\ \textgreater \ -35 \text { and } -5 x \geq-50[/tex]
Answer (1)
Solve the first inequality -35"> 7 x > − 35 by dividing both sides by 7, resulting in -5"> x > − 5 .
Solve the second inequality − 5 x ≥ − 50 by dividing both sides by -5 (and flipping the inequality sign), resulting in x ≤ 10 .
Combine the two inequalities to find the intersection: − 5 < x ≤ 10 .
The solution to the compound inequality is − 5 < x ≤ 10 .
Explanation
Understanding the Problem We are given the compound inequality -35"> 7 x > − 35 and − 5 x ≥ − 50 . We need to solve each inequality separately and then find the intersection of their solution sets. This will give us the solution to the compound inequality.
Solving the First Inequality Let's solve the first inequality, -35"> 7 x > − 35 . To isolate x , we divide both sides of the inequality by 7. Since 7 is a positive number, we don't need to flip the inequality sign. We get: \frac{-35}{7}"> 7 7 x > 7 − 35 -5"> x > − 5
Solving the Second Inequality Now, let's solve the second inequality, − 5 x ≥ − 50 . To isolate x , we divide both sides of the inequality by -5. Since we are dividing by a negative number, we need to flip the inequality sign. We get: − 5 − 5 x ≤ − 5 − 50 x ≤ 10
Finding the Intersection We need to find the values of x that satisfy both -5"> x > − 5 and x ≤ 10 . This means x must be greater than -5 and less than or equal to 10. We can write this as a compound inequality: − 5 < x ≤ 10
Final Answer The solution to the compound inequality is − 5 < x ≤ 10 . This means x can be any number greater than -5, but it must be less than or equal to 10.
Examples
Compound inequalities are useful in many real-world situations. For example, suppose a company wants to hire workers who are at least 18 years old but no older than 65 years old. This can be expressed as a compound inequality: 18 ≤ x ≤ 65 , where x is the age of the worker. Another example is when designing a bridge, engineers need to ensure that the bridge can withstand a certain range of temperatures. This can be expressed as a compound inequality, where x is the temperature.
