Verified Answer :

  • Isolate the variable by subtracting \(7x\) from both sides and adding \(12\) to both sides: \(3x - 7x \geq 4 + 12\), simplifying to \(-4x \geq 16\).
  • Divide by \(-4\), remembering to flip the inequality sign: \(x \leq -4\).
  • The solution is the interval of all real numbers less than or equal to \(-4\).
  • Graph this on a number line using a closed point at \(-4\) and a ray pointing to the left: \(\boxed{x \leq -4}\).

Explanation

  1. Analysis and Objective

The objective is to find the set of values for \(x\) that satisfy the inequality \(3x - 12 \geq 7x + 4\). This is a linear inequality, and our goal is to isolate \(x\) on one side to determine its range. We need to remember that dividing or multiplying both sides of an inequality by a negative number reverses the direction of the inequality sign. Finally, we will represent this solution graphically on a number line, which is a standard way to visualize the solution set for linear inequalities in one variable.

  1. Isolating the Variable Term

To solve the inequality \(3x - 12 \geq 7x + 4\), we first collect all terms involving \(x\) on one side and the constant terms on the other. Subtracting \(7x\) from both sides gives:
\[3x - 7x - 12 \geq 4\]
Next, add \(12\) to both sides of the inequality to isolate the term with \(x\):
\[3x - 7x \geq 4 + 12\]
Simplifying both sides yields:
\[-4x \geq 16\]

  1. Solving for x

Now, we divide both sides by \(-4\). Since we are dividing by a negative number, the direction of the inequality symbol must be reversed from \(\geq\) to \(\leq\).
\[x \leq \frac{16}{-4}\]
\[x \leq -4\]
This gives us the solution set as all real numbers less than or equal to \(-4\).

  1. Graphing the Solution

To graph the solution \(x \leq -4\) on a number line, we identify the boundary point at \(-4\). Since the inequality is "less than or equal to" (\(\leq\)), the value \(-4\) is included in the solution set, which we represent by marking a closed point at \(-4\). Because we want all values less than \(-4\), we draw a ray starting from this point and extending to the left.
The final solution is \(x \leq -4\). Thus, the solution set consists of all values \(\boxed{x \leq -4}\).

Examples

Suppose you are managing a budget for a project and the costs are rising. You have a constraint that your current spending, represented by \(3x - 12\), must be greater than or equal to a minimum target project requirement, which is \(7x + 4\). Solving this inequality helps you determine the range of values for a variable (like unit cost or hours worked) that maintains your budget constraint. By understanding these thresholds, you can make informed decisions to ensure your project remains within its necessary operational limits.

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