Graph the solution set to this inequality.
[tex]3 x-11\ \textgreater \ 7 x+9[/tex]
Answer (1)
Subtract 3 x from both sides: 4x + 9"> − 11 > 4 x + 9 .
Subtract 9 from both sides: 4x"> − 20 > 4 x .
Divide both sides by 4: x"> − 5 > x , which is equivalent to x < − 5 .
The solution set is all real numbers less than -5, which is graphed with an open circle at -5 and shading to the left. x < − 5
Explanation
Understanding the Problem We are given the inequality 7x + 9"> 3 x − 11 > 7 x + 9 . Our goal is to solve for x and then describe how to graph the solution set on a number line.
Subtracting 3 x from Both Sides First, we want to isolate the variable x on one side of the inequality. To do this, we can subtract 3 x from both sides of the inequality:
7x + 9 - 3x"> 3 x − 11 − 3 x > 7 x + 9 − 3 x
This simplifies to:
4x + 9"> − 11 > 4 x + 9
Subtracting 9 from Both Sides Next, we subtract 9 from both sides of the inequality:
4x + 9 - 9"> − 11 − 9 > 4 x + 9 − 9
This simplifies to:
4x"> − 20 > 4 x
Dividing Both Sides by 4 Now, we divide both sides of the inequality by 4:
\frac{4x}{4}"> 4 − 20 > 4 4 x
This simplifies to:
x"> − 5 > x
This is equivalent to x < − 5 .
Graphing the Solution Set The solution set is all real numbers less than -5. To graph this solution set on a number line, we draw an open circle at -5 to indicate that -5 is not included in the solution set, and then we shade the line to the left of -5 to represent all numbers less than -5.
Final Answer The solution to the inequality 7x + 9"> 3 x − 11 > 7 x + 9 is x < − 5 . This is graphed on a number line with an open circle at -5 and the line shaded to the left.
Examples
Imagine you're setting up a budget where you want to ensure your expenses ( 3 x − 11 ) are always less than a certain limit ( 7 x + 9 ). Solving this inequality helps you determine the maximum value of x (e.g., hours worked, units consumed) to stay within your budget. Graphing the solution set visually represents all possible values of x that satisfy your budget constraint, making it easy to see the range of acceptable values. This type of problem is useful in personal finance, resource management, and any situation where you need to keep one quantity below another.
