Use the drawing tool(s) to form the correct answer on the number line. Graph the solution to this inequality on the number line: $-4x + 7 > x - 13$
Answer (1)
Add 4 x to both sides: 5x - 13"> 7 > 5 x − 13 .
Add 13 to both sides: 5x"> 20 > 5 x .
Divide both sides by 5 : x"> 4 > x .
The solution is x < 4 , represented on the number line with an open circle at 4 and an arrow extending to the left. x < 4
Explanation
Understanding the Problem We are given the inequality x - 13"> − 4 x + 7 > x − 13 . Our goal is to solve for x and represent the solution on a number line.
Adding 4 x to Both Sides First, we want to isolate x on one side of the inequality. To do this, we can add 4 x to both sides of the inequality: x - 13 + 4x"> − 4 x + 7 + 4 x > x − 13 + 4 x This simplifies to: 5x - 13"> 7 > 5 x − 13
Adding 13 to Both Sides Next, we add 13 to both sides of the inequality: 5x - 13 + 13"> 7 + 13 > 5 x − 13 + 13 This simplifies to: 5x"> 20 > 5 x
Dividing by 5 Now, we divide both sides of the inequality by 5 : \frac{5x}{5}"> 5 20 > 5 5 x This simplifies to: x"> 4 > x This is equivalent to x < 4 .
Representing on the Number Line The solution to the inequality is x < 4 . To represent this on the number line, we use an open point at 4 (since x is strictly less than 4 , not less than or equal to) and a ray extending to the left, indicating all values less than 4 .
Final Answer The solution to the inequality x - 13"> − 4 x + 7 > x − 13 is x < 4 . This is represented on the number line with an open circle at 4 and an arrow extending to the left.
Examples
Imagine you're setting a budget for a project. You have a constraint that your expenses, represented by − 4 x + 7 , must be greater than the estimated income, represented by x − 13 . Solving the inequality helps you determine the maximum value of x (e.g., hours worked, materials used) to ensure your expenses stay within the budget. Understanding and solving such inequalities is crucial in managing resources and making informed financial decisions in various real-life scenarios.
