Problem 39 - Algebra for College Students

Question

Solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. \(2 x-1 \geq 5 \quad\) and \(\quad x>0\)

Step-by-Step Solution

Answer

The solution set is \([3, \infty)\).

Step 1: Solve the first inequality

Begin by solving the inequality \(2x - 1 \geq 5\). To isolate \(x\), first add 1 to both sides of the inequality: \(2x - 1 + 1 \geq 5 + 1\), which simplifies to \(2x \geq 6\). Next, divide each side of this inequality by 2 to solve for \(x\): \(x \geq 3\).

Step 2: Solve the second inequality

The second inequality is \(x > 0\). This inequality is already simplified, so we just note it as part of our solution.

Step 3: Determine the combined solution

Since we need \(x\) to satisfy both inequalities, we look for an \(x\) that is both greater than or equal to 3, and also greater than 0. Since \(x \geq 3\) already satisfies \(x > 0\), the combined solution can be written as \(x \geq 3\).

Step 4: Express the solution in interval notation

The solution set for \(x \geq 3\) is expressed in interval notation as \([3, \infty)\). This indicates all numbers from 3 to infinity, including 3.

Step 5: Graph the solution set

To graph the solution \([3, \infty)\), draw a number line and place a closed circle at 3 (indicating that 3 is included in the solution set). Shade the region to the right of 3, extending towards positive infinity, to represent all numbers greater than or equal to 3.

Key Concepts

Inequality Solving

When solving compound inequalities, the goal is to find a set of values that satisfy all given inequalities. Let's consider each inequality separately and then combine the solutions.
First, take the inequality \(2x - 1 \geq 5\). To isolate \(x\), start by adding 1 to both sides. This yields \(2x \geq 6\). Next, divide both sides by 2 to solve for \(x\), resulting in \(x \geq 3\).
The second inequality, \(x > 0\), is straightforward since it's already in its simplest form. There's no need for further actions to solve it.
In compound inequalities joined by "and," the solution is any value of \(x\) that satisfies both conditions simultaneously. Here, since both conditions are satisfied by \(x \geq 3\), \(x \geq 3\) is the combined solution for both inequalities.

Interval Notation

Once we solve an inequality, we represent the solution set using interval notation, which offers a concise way to express ranges of values.
For the solution \(x \geq 3\), interval notation is \([3, \infty)\). This means:\

• The bracket \([\) means that the number 3 is included in the solution set.
• The comma inside the brackets separates the lower boundary (3) from the description of the unbounded upper range.
• The parenthesis \()\) next to infinity means infinity is not inclusive, illustrating that the values extend indefinitely.

Understanding how to write this notation is essential when solving inequalities because it gives a clear picture of all possible solutions. So always ensure the boundaries are correctly set, whether numbers are included or not, through appropriate use of brackets or parentheses.

Graphing Solutions

Graphing is another useful tool to visually represent the solutions of inequalities, helping to easily understand which values are included.
For \([3, \infty)\), begin by drawing a horizontal number line. At 3, place a closed circle. This indicates that 3 is part of the solution.
Next, shade the entire region to the right of 3 extending indefinitely. This represents all numbers greater than or equal to 3.

• A closed circle signifies inclusion of that value in the solution set.
• Shading to the right of a point represents values greater than that point.
• An arrow or extension line shows the indefinite nature of extending towards infinity.

Graphing such solutions provides a clear and simple visual understanding of the values included and how far the solution extends.