Verified Answer :

  • Distribute the constant on the left side: \(-3x - 2 \leq -2x + 11\).
  • Move variable terms to one side: \(-2 \leq x + 11\).
  • Isolate the variable: \(x \geq -13\).
  • The correct graph is a number line with a closed circle at \(-13\) and a shaded line pointing to the right, representing \(\boxed{x \geq -13}\).

Explanation

  1. Simplifying the inequality

To solve the inequality \(-\frac{1}{4}(12x + 8) \leq -2x + 11\), we first simplify the expression on the left-hand side. Distributing the \(-\frac{1}{4}\) into the parentheses gives:
\[-\frac{1}{4} \cdot 12x - \frac{1}{4} \cdot 8 = -3x - 2\]
Thus, the inequality becomes:
\[-3x - 2 \leq -2x + 11\]

  1. Isolating the variable x

Next, we isolate the variable \(x\). We can do this by adding \(3x\) to both sides of the inequality:
\[-2 \leq x + 11\]
Now, subtract \(11\) from both sides to solve for \(x\):
\[-2 - 11 \leq x\]
\[-13 \leq x\]
This is equivalent to:
\[x \geq -13\]

  1. Interpreting the solution graphically

The solution \(x \geq -13\) represents all real numbers greater than or equal to \(-13\). On a number line, this is represented by a closed circle at the point \(-13\) to indicate that it is included in the solution, and a solid ray extending to the right, representing all values greater than \(-13\).

Examples

Imagine you are budgeting for a project and you have a base cost of 11 dollars and you earn 2 dollars for every hour you work. If you have to pay a penalty of \(\frac{1}{4}\) of your total initial work load (12 hours + 8 extra units of labor), you need to ensure your total income covers your remaining costs. Solving inequalities like this helps determine the minimum amount of time or resources required to meet a specific goal. In this case, finding \(x \geq -13\) helps determine the threshold needed to maintain a balanced budget.

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