Problem 46 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$\frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12}$$

Step-by-Step Solution

Answer

The solution of the given inequality \(x \geq \frac{-19}{7}\) can be expressed as [\(-\frac{19}{7}, +\infty)\] in interval notation. The graph of this solution on a number line starts at \(-\frac{19}{7}\) and extends to the right to indicate all numbers up to and including +\infty.

Step 1: Simplify the Inequality

First, we need to multiply the entire inequality by the LCM (Least Common Multiple) of 6 and 12, which is 12. This simplifies the equation as follows:\n \(8x - 6 + 24 \geq x - 1\). Then simplify to get \(7x \geq -19\).

Step 2: Solve for the Variable

Next, isolate x by dividing both sides of the equation by 7. We get \(x \geq \frac{-19}{7}\).

Step 3: Write the Solution in Interval Notation

In interval notation, the solution set is represented as [\(-\frac{19}{7}, +\infty)\]. This represents all x-values that are greater than or equal to \(-\frac{19}{7}\) and less than +\infty.

Step 4: Graph the Solution Set

On the number line, we graph this solution set as a closed circle at \(-\frac{19}{7}\) with an arrow extending to the right, indicating all numbers up to and including \(\infty\). The closed circle indicates that \(-\frac{19}{7}\) is included in the solution set (because of the 'greater than or equal to' symbol in the original inequality.)