Solve $-4 x-6<-18$ and $7-12 x>-65$
Answer (1)
Solve the first inequality − 4 x − 6 < − 18 to get 3"> x > 3 .
Solve the second inequality -65"> 7 − 12 x > − 65 to get x < 6 .
Find the intersection of the two solution sets: 3 < x < 6 .
Express the solution as an interval: 3 < x < 6 .
Explanation
Understanding the Problem We are given two inequalities: − 4 x − 6 < − 18 and -65"> 7 − 12 x > − 65 . Our goal is to find all values of x that satisfy both inequalities simultaneously. This means we need to solve each inequality separately and then find the intersection of their solution sets.
Solving the First Inequality Let's solve the first inequality, − 4 x − 6 < − 18 . To isolate x , we first add 6 to both sides of the inequality: − 4 x − 6 + 6 < − 18 + 6
− 4 x < − 12
Now, we divide both sides by -4. Remember that when we divide or multiply an inequality by a negative number, we must reverse the inequality sign: \frac{-12}{-4}"> − 4 − 4 x > − 4 − 12
3"> x > 3
So, the solution to the first inequality is 3"> x > 3 .
Solving the Second Inequality Next, let's solve the second inequality, -65"> 7 − 12 x > − 65 . First, we subtract 7 from both sides: -65 - 7"> 7 − 12 x − 7 > − 65 − 7
-72"> − 12 x > − 72
Now, we divide both sides by -12. Again, we must reverse the inequality sign because we are dividing by a negative number: − 12 − 12 x < − 12 − 72
x < 6
So, the solution to the second inequality is x < 6 .
Finding the Intersection Now we need to find the values of x that satisfy both 3"> x > 3 and x < 6 . This means x must be greater than 3 and less than 6. We can write this as a compound inequality: 3 < x < 6
In interval notation, this is the interval ( 3 , 6 ) .
Final Answer Therefore, the solution to the compound inequality − 4 x − 6 < − 18 and -65"> 7 − 12 x > − 65 is 3 < x < 6 .
Examples
Imagine you're planning a surprise party and need to keep the number of guests within a certain range. One condition requires more than 3 friends to make it lively, while another limits the group to fewer than 6 to keep it manageable in your apartment. Solving the compound inequality helps you determine the possible number of guests, ensuring the party is both fun and feasible. This type of problem is also useful in setting constraints in optimization problems, such as determining the range of values for a variable in a linear programming problem.
