Which graph represents the solution to the inequality $-4m - 3 > 9$?
Answer (2)
Add 3 to both sides: 12"> − 4 m > 12 .
Divide both sides by -4 (and flip the inequality sign): m < − 3 .
The solution is all numbers less than -3.
Graphically, this is an open circle at -3 with a line extending to the left: m < − 3 .
Explanation
Understanding the Problem We are given the inequality 9"> − 4 m − 3 > 9 and we want to find which graph represents its solution.
Isolating the Variable First, we need to isolate the variable m . To do this, we add 3 to both sides of the inequality: 9 + 3"> − 4 m − 3 + 3 > 9 + 3
Simplifying the Inequality This simplifies to: 12"> − 4 m > 12
Dividing by a Negative Number Now, we divide both sides by -4. Remember that when we divide or multiply an inequality by a negative number, we must flip the inequality sign: − 4 − 4 m < − 4 12
Finding the Solution This simplifies to: m < − 3
Graphical Representation The solution to the inequality is m < − 3 . This means that m can be any number less than -3, but it cannot be equal to -3. On a number line, this is represented by an open circle at -3 and a line extending to the left, indicating all values less than -3.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're managing a budget and need to ensure your expenses ( e ) stay below a certain limit, say 1000. T hi sc anb ee x p resse d a s e < 1000$. Solving and graphing inequalities helps you visualize and manage such constraints effectively, ensuring you don't exceed your financial limits. Similarly, in science, inequalities are used to define acceptable ranges for experimental conditions, ensuring results remain valid and reliable.
The solution to the inequality 9"> − 4 m − 3 > 9 is m < − 3 , which means all values less than − 3 are valid. This is represented graphically by an open circle at − 3 and a line extending to the left on a number line. Therefore, any graph showing this representation correctly depicts the solution.
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