Solve the inequality $3x > 5x + 8$
Answer (1)
Subtract 5 x from both sides: 8"> − 2 x > 8 .
Divide both sides by − 2 , remembering to flip the inequality sign: x < − 4 .
The solution to the inequality is x < − 4 .
Therefore, the solution is x < − 4 .
Explanation
Understanding the Inequality We are given the inequality 5x + 8"> 3 x > 5 x + 8 . Our goal is to isolate x on one side of the inequality to find the solution.
Isolating the x Term First, let's subtract 5 x from both sides of the inequality to group the x terms together:
5x + 8 - 5x"> 3 x − 5 x > 5 x + 8 − 5 x
This simplifies to:
8"> − 2 x > 8
Dividing by a Negative Number Now, we need to divide both sides of the inequality by − 2 to solve for x . Remember that when we divide (or multiply) an inequality by a negative number, we must reverse the direction of the inequality sign.
− 2 − 2 x < − 2 8
The Solution Simplifying the inequality, we get:
x < − 4
This means that any value of x that is less than − 4 will satisfy the original inequality.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might need to ensure that your expenses ( x ) are less than your income. If your income is represented by the expression 5 x + 8 and your expenses are 3 x , solving the inequality 5x + 8"> 3 x > 5 x + 8 helps you determine the values of x for which your expenses exceed your income, indicating a potential budget deficit. This kind of problem-solving is also applicable in business, engineering, and other fields where constraints and limits need to be analyzed.
