Verified Answer :

  • Simplify the inequality to \( -2x - 2 > 8 \).
  • Isolate the variable to obtain \( x < -5 \).
  • Compare the provided options against the solution \( x < -5 \).
  • Identify that the valid solutions are \(\boxed{-9, -7}\).

Explanation

  1. Problem Analysis

We are given the inequality:
\[3x - 2 - 5x > 5x + 8 - 5x\]
Our goal is to simplify this expression to isolate \(x\) and determine the range of values that satisfy it, then select the appropriate options from the set \(\{-5, 1, -9, 5, -7, 3\}\).

  1. Simplifying the Inequality

To solve the inequality, we first simplify both sides:

  • Left-hand side: \(3x - 2 - 5x = (3 - 5)x - 2 = -2x - 2\)
  • Right-hand side: \(5x + 8 - 5x = (5 - 5)x + 8 = 8\)

So the inequality becomes:
\[-2x - 2 > 8\]

  1. Solving for x

Next, we isolate the term with \(x\):

  • Add 2 to both sides of the inequality:

\[-2x > 8 + 2\]
\[-2x > 10\]

  • Divide both sides by -2. Remember that when dividing or multiplying an inequality by a negative number, the inequality sign must be reversed:

\[x < \frac{10}{-2}\]
\[x < -5\]

  1. Selecting the Valid Solutions

We must check which of the given options satisfy \(x < -5\):

  • For \(x = -5\): \(-5 < -5\) is False.
  • For \(x = 1\): \(1 < -5\) is False.
  • For \(x = -9\): \(-9 < -5\) is True.
  • For \(x = 5\): \(5 < -5\) is False.
  • For \(x = -7\): \(-7 < -5\) is True.
  • For \(x = 3\): \(3 < -5\) is False.

The valid solutions are \(-9\) and \(-7\).

Examples

In financial planning, setting a budget involves comparing income against expenses. Suppose you have a fluctuating daily budget represented by an algebraic expression, and you need to ensure your total daily expenditure stays below a specific limit to maintain savings. This problem models such a situation, where solving the inequality helps you identify which spending habits (represented by variable values) meet your financial constraints, ensuring your account remains in the positive range.

Related Questions