Solve $-3 x-5<-20$ and $8-11 x>-80$
Answer (1)
Solve the first inequality − 3 x − 5 < − 20 and get 5"> x > 5 .
Solve the second inequality -80"> 8 − 11 x > − 80 and get x < 8 .
Find the intersection of the two solution sets, which is 5 < x < 8 .
The solution to the compound inequality is 5 < x < 8 , which means x is greater than 5 and less than 8. 5 < x < 8
Explanation
Understanding the Problem We are given two inequalities: − 3 x − 5 < − 20 and -80"> 8 − 11 x > − 80 . We need to find the values of x that satisfy both inequalities. The solution will be the intersection of the solution sets of the two inequalities.
Solving the First Inequality Let's solve the first inequality: − 3 x − 5 < − 20 . Add 5 to both sides: − 3 x < − 20 + 5 − 3 x < − 15 Divide both sides by -3. Remember to flip the inequality sign when dividing by a negative number: \frac{-15}{-3}"> x > − 3 − 15 5"> x > 5
Solving the Second Inequality Now, let's solve the second inequality: -80"> 8 − 11 x > − 80 . Subtract 8 from both sides: -80 - 8"> − 11 x > − 80 − 8 -88"> − 11 x > − 88 Divide both sides by -11. Remember to flip the inequality sign when dividing by a negative number: x < − 11 − 88 x < 8
Finding the Intersection We need to find the values of x that satisfy both 5"> x > 5 and x < 8 . This means x must be greater than 5 and less than 8. In interval notation, this is ( 5 , 8 ) .
Final Answer Therefore, the solution to the compound inequality is 5 < x < 8 .
Examples
Imagine you're planning a surprise party and need to invite more than 5 but fewer than 8 friends to fit comfortably in your living room. Solving inequalities like this helps determine the possible number of guests, ensuring the party is neither too crowded nor too empty. This concept is useful in resource allocation, capacity planning, and setting boundaries in various real-life situations.
