Problem 41 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution of the given inequality is \(x \geq 2\), represented in interval notation as \([2, +\infty)\).

Step 1: Rearrange the inequality

First, isolate the term containing the variable by subtracting \(\frac{x}{2}\) from both sides. This gives \(\frac{x}{4}-\frac{x}{2} \leq 1 - \frac{3}{2}\). Simplify it further to \(-\frac{x}{4} \leq -\frac{1}{2}\).

Step 2: Solve for x

Then, solve for x by multiplying both sides by -4 (recall that inequalities switch the direction when multiplied or divided by any negative number). This will give \(x \geq 2\).

Step 3: Represent solution in interval notation

The solution set in interval notation is \([2, +\infty)\). This denotes that x can be any real number, equal to or greater than 2.

Step 4: Plot the solution on a number line

Lastly, plot the solution on a number line, including the point at x=2 (represented by a closed circle or a bracket), with all points to the right of this point included in the solution set since the inequality is 'greater than or equal to'.