Solve the inequality and graph the solution.
[tex]9 m+1<33+4(6 m-11)-18 m[/tex]
Answer (2)
Expand the right side of the inequality: 9 m + 1 < 33 + 24 m − 44 − 18 m .
Simplify the inequality: 9 m + 1 < − 11 + 6 m .
Isolate m by subtracting 6 m and 1 from both sides: 3 m < − 12 .
Solve for m : m < − 4 . The solution in interval notation is ( − ∞ , − 4 ) .
Explanation
Initial Inequality We are given the inequality 9 m + 1 < 33 + 4 ( 6 m − 11 ) − 18 m
Expanding the Right Side First, we need to expand the right side of the inequality: 9 m + 1 < 33 + 24 m − 44 − 18 m
Simplifying the Inequality Now, we simplify the right side by combining like terms: 9 m + 1 < 33 − 44 + 24 m − 18 m
9 m + 1 < − 11 + 6 m
Isolating the Variable Next, we want to isolate the variable m on one side of the inequality. We can subtract 6 m from both sides: 9 m − 6 m + 1 < − 11 + 6 m − 6 m
3 m + 1 < − 11
Further Isolation Now, subtract 1 from both sides: 3 m + 1 − 1 < − 11 − 1
3 m < − 12
Solving for m Finally, we divide both sides by 3 to solve for m :
3 3 m < 3 − 12
m < − 4
Interval Notation The solution to the inequality is m < − 4 . In interval notation, this is ( − ∞ , − 4 ) .
Graphing the Solution To graph the solution on a number line, we draw an open circle at − 4 (since m is strictly less than − 4 ) and shade the region to the left of − 4 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're managing a budget and need to ensure your expenses ( m ) stay below a certain limit. If your limit is egative 4000 , t h e in e q u a l i t y m < -4000$ helps you visualize and manage your spending to stay within your financial constraints. Similarly, in manufacturing, tolerances are often expressed as inequalities to ensure product quality.
The solution to the inequality 9 m + 1 < 33 + 4 ( 6 m − 11 ) − 18 m is m < − 4 , expressed in interval notation as ( − ∞ , − 4 ) . To graph this on a number line, you mark an open circle at − 4 and shade to the left. This solution helps illustrate concepts in real-world budgeting and limits.
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