Verified Answer :

  • Distribute the constant -3 across the term \((x + 2)\) to get \(x - 3x - 6 > 4\).
  • Simplify the left side by combining like terms to obtain \(-2x - 6 > 4\).
  • Add 6 to both sides and divide by -2, remembering to flip the inequality sign.
  • Conclude that the solution is \(x < -5\), which is represented by an open circle at -5 with the region to the left shaded on a number line.
  • The final answer is \(\boxed{x < -5}\).

Explanation

  1. Distributing and Simplifying the Inequality

The goal is to solve the linear inequality \(x - 3(x + 2) > 4\). To begin, we need to simplify the expression by applying the distributive property to the term \(-3(x + 2)\). This involves multiplying -3 by both \(x\) and \(2\). We have:
\[x - 3x - 6 > 4\]
Next, we combine the like terms on the left side. The expression \(x - 3x\) simplifies to \(-2x\), giving us:
\[-2x - 6 > 4\]

  1. Isolating the Variable and Final Result

Now, we isolate the variable term \(-2x\) by moving the constant term \(-6\) to the right side of the inequality. We do this by adding 6 to both sides:
\[-2x > 4 + 6\]
\[-2x > 10\]
Finally, to solve for \(x\), we divide both sides by -2. A fundamental rule of inequalities is that when dividing or multiplying by a negative number, the direction of the inequality sign must be reversed. Thus, the greater-than sign (\(>\)) becomes a less-than sign (\(<\)):
\[x < \frac{10}{-2}\]
\[x < -5\]
Therefore, the inequality is satisfied for all values of \(x\) strictly less than -5.

Examples

Inequalities are essential in budgeting. Imagine you are planning a small project with a total budget, but you have fixed initial costs. If \(x\) represents your potential variable spending, you might use an inequality to ensure your total expenditure stays below a certain limit. For example, if you have to pay a fixed fee and a per-unit cost, solving an inequality helps you determine the maximum number of units you can afford while remaining within your financial constraints. This keeps your project running smoothly without overspending.

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