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Chapter 2: Problem 42

Solve each inequality and express the solution set using interval notation. 3 x-2<12

Short Answer

Expert verified
(-鈭�, 14/3)

Step by step solution

01

Add 2 to both sides

Start with the inequality \(3x - 2 < 12\). Add 2 to both sides to isolate the term with \(x\) on one side: \[3x - 2 + 2 < 12 + 2\] which simplifies to \(3x < 14\).
02

Divide by 3

Now that we have \(3x < 14\), divide both sides by 3 to solve for \(x\): \[x < \frac{14}{3}\].
03

Express the solution in interval notation

The solution \(x < \frac{14}{3}\) can be expressed in interval notation as \((-\infty, \frac{14}{3})\). This represents all values less than \(\frac{14}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Solution Steps
Solving inequalities involves a few methodical steps to find the values that satisfy the inequality. Let's take the inequality \(3x - 2 < 12\) as an example.
  • Step 1: Isolate the variable term.
    Begin by getting \(x\) by itself on one side of the inequality sign. To do this, add 2 to both sides of the inequality: \[ 3x - 2 + 2 < 12 + 2 \] This will give us: \(3x < 14\).

  • Step 2: Solve for the variable.
    Once the term with \(x\) is isolated, divide each side by the coefficient of \(x\), which is 3 in this case. The inequality becomes: \[ x < \frac{14}{3} \]
By following these steps, you successfully isolate \(x\) and identify the range of values that satisfy the original inequality.
Expressing Solution Sets
After finding the variable's range, the next step is to clearly express the solution set. A solution set includes all possible values of \(x\) that make the inequality true.
For our example, the solution \(x < \frac{14}{3}\) means that \(x\) can be any number smaller than \(\frac{14}{3}\). There is no specific number that \(x\) must equal, rather it could be anything within a specific range.
  • When expressing a solution set, ensure you understand whether the boundary number (in this case, \(\frac{14}{3}\)) is included or not. Since the inequality is "less than" and not "less than or equal to", \(\frac{14}{3}\) is not included in the solution set.
Understanding how to express solution sets accurately will help you identify the complete scope of possible values for \(x\).
Interval Notation
Interval notation is a concise way to represent a range of numbers that make an inequality true.
For the inequality \(x < \frac{14}{3}\), the interval notation is written as \(( -\infty, \frac{14}{3} )\).
  • The round bracket \((\) opening the interval indicates that \(-\infty\) (infinity itself) cannot be reached.
  • The round bracket \() )\) closing after \(\frac{14}{3}\) tells us that \(\frac{14}{3}\) is not included in the interval.
Interval notation provides a straightforward and universally understood form to describe solution sets. It helps visualize where the set starts and ends, simplifying understanding of the values \(x\) can take. Using this form, we quickly understand that any number less than \(\frac{14}{3}\) is part of our solution.

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