Verified Answer :

  • Start by adding 2 to both sides of the inequality \(-2 + 6x \geq 34\) to get \(6x \geq 36\).
  • Divide both sides by 6 to isolate \(x\), resulting in \(x \geq 6\).
  • The solution set represents all real numbers \(x\) greater than or equal to 6.
  • Final answer: \(\boxed{x \geq 6}\)

Explanation

  1. Problem Analysis

We are given the linear inequality \(-2 + 6x \geq 34\). Our objective is to find the set of all real numbers \(x\) that satisfy this inequality. A linear inequality represents a relationship where one expression is greater than or equal to another, and we can manipulate it similarly to linear equations, provided we follow the rules of inequalities (like reversing the sign when multiplying or dividing by a negative number). Here, our goal is to isolate the variable \(x\).

  1. Isolating the term with \(x\): Adding 2 to both sides

To isolate the term containing \(x\), we first add 2 to both sides of the inequality. This operation maintains the direction of the inequality:
\[-2 + 6x + 2 \geq 34 + 2\]
Simplifying this, we get:
\[6x \geq 36\]

  1. Solving for \(x\): Dividing by 6

Now, to solve for \(x\), we divide both sides by the coefficient of \(x\), which is 6. Since 6 is a positive number, the direction of the inequality remains unchanged:
\[\frac{6x}{6} \geq \frac{36}{6}\]
Performing the division, we obtain:
\[x \geq 6\]
Thus, any real number greater than or equal to 6 satisfies the original inequality.

Examples

Imagine you are setting up a budget for an event where there is a fixed rental fee reduction of \$2 (representing the -2 in our equation) and a hourly rate for service of \$6 per hour. If you have a total of at least \$34 to cover the cost, you want to know the minimum number of hours (\(x\)) you can afford. By solving \(-2 + 6x \geq 34\), you find that you need to work for at least 6 hours to meet or exceed your budget requirement.

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