Solve the compound inequality below.

[tex]6+7 m\ \textless \ 6 m-5[/tex] or [tex]3 m-7\ \textless \ 5+6 m[/tex]

Solve the first inequality: 6 + 7 m < 6 m − 5 which simplifies to m < − 11 .
Solve the second inequality: 3 m − 7 < 5 + 6 m which simplifies to -4"> m > − 4 .
Combine the solutions using 'or': m < − 11 or -4"> m > − 4 .
The solution is m < − 11 or -4"> m > − 4 , representing all numbers less than -11 or greater than -4: -4}"> m < − 11 or m > − 4 ​ .

Explanation

Understanding the Problem We are given the compound inequality: 6 + 7 m < 6 m − 5 or 3 m − 7 < 5 + 6 m . We need to solve for m .

Solving the First Inequality First, let's solve the inequality 6 + 7 m < 6 m − 5 . Subtracting 6 m from both sides gives 6 + m < − 5 . Subtracting 6 from both sides gives m < − 11 .

Solving the Second Inequality Next, let's solve the inequality 3 m − 7 < 5 + 6 m . Subtracting 3 m from both sides gives − 7 < 5 + 3 m . Subtracting 5 from both sides gives − 12 < 3 m . Dividing both sides by 3 gives − 4 < m , which is the same as -4"> m > − 4 .

Combining the Solutions The compound inequality is m < − 11 or -4"> m > − 4 . This means m can be any number less than -11 or any number greater than -4.


Examples
Compound inequalities are useful in many real-world scenarios. For example, a company might set a sales target such that to get a bonus, a salesperson must sell either less than 10 units (probation) or more than 50 units (top performer). Similarly, in manufacturing, a machine might be considered to be functioning correctly if a measurement is within a certain range, or outside another range indicating a malfunction.

Answered by GinnyAnswer | 2025-07-03

The solution to the compound inequality is m < − 11 or -4"> m > − 4 , meaning m can be any number less than − 11 or greater than − 4 .
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Answered by Anonymous | 2025-07-04