Problem 77 - Intermediate Algebra

Question

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the compound inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. \(3 x-2 \geq 4\) and \(x+6 \geq 12\) b. \(3 x-2 \geq 4\) or \(x+6 \geq 12\)

Step-by-Step Solution

Answer

a: \([6, \infty)\); b: \([2, \infty)\).

Step 1: Solve the First Inequality

First, solve the inequality \(3x - 2 \geq 4\). Add 2 to both sides to get \(3x \geq 6\). Then divide both sides by 3 to find \(x \geq 2\).

Step 2: Solve the Second Inequality

Next, solve the inequality \(x + 6 \geq 12\). Subtract 6 from both sides to get \(x \geq 6\).

Step 3: Determine the Solution Set for the Compound Inequality (Part a)

Since the compound inequality in part a uses 'and', we take the intersection of the two solutions found. Since \(x \geq 2\) and \(x \geq 6\), the solution is simply \(x \geq 6\).

Step 4: Write Interval Notation for Part a

The solution set for part a in interval notation is \([6, \infty)\).

Step 5: Compound Inequality Solution Set for Part b

For part b, the compound inequality uses 'or'. Therefore, we take the union of the solutions. \(x \geq 2\) or \(x \geq 6\) gives the result \(x \geq 2\).

Step 6: Write Interval Notation for Part b

The solution set for part b in interval notation is \([2, \infty)\).

Key Concepts

Compound Inequality

A compound inequality is a mathematical statement that involves two or more inequalities joined by the word "and" or "or". When we use "and", it means that both conditions must be true simultaneously, which typically results in finding the intersection of solutions. In contrast, using "or" means that any one of the conditions can be true, leading us to find the union of solutions.

In the original exercise, part a uses "and", indicating that both inequalities must be satisfied. So, the solution is only those values of \(x\) that are true for both conditions. For part b, "or" is used, so any \(x\) that satisfies either inequality will be a part of the solution set.

• "And" – Requires both conditions to be true.
• "Or" – Only one condition needs to be true.

Interval Notation

Interval notation is a mathematical shorthand used to represent a set of numbers. This notation uses brackets to indicate which numbers are included in a set.

When writing interval notation:

• Use \[ \] to include a boundary in the solution set.
• Use \( ( \) or \( ) \) to exclude a boundary.

For example, in part a, we wrote the solution \(x \geq 6\) as the interval \[6, \infty)\]. The square bracket includes the number 6, while the parenthesis on infinity signifies that infinity is not a number we reach. Similarly, for part b, the solution \(x \geq 2\) becomes \[2, \infty)\]. This format concisely captures infinite ranges and boundary inclusions.

Solution Set

The solution set is the collection of all possible solutions to an inequality or equation.

When finding the solution set for a compound inequality, it's essential to determine the type of connection—"and" or "or"—as this affects how we combine individual solutions. For instance, in part a, that used "and", required us to find where both conditions overlap. Consequently, the solution set was \(x \geq 6\).

In part b, with "or", any solution that met at least one condition was included, leading to the broader solution set of \(x \geq 2\). Understanding the logic behind "and" and "or" is crucial in determining the correct solutions.

Graphing Inequalities

Graphing inequalities involves visually representing the solution set on a number line. This visual can be especially helpful for understanding how many types of solutions work, such as intersections and unions.

To graph, first solve the inequality. For example, with \(x \geq 2\), you'd draw an arrow starting at 2 and extending to the right, since all values greater than or equal to 2 are included. Use a solid circle on the number to indicate it's part of the solution.

When dealing with "and", only the overlapping regions of two graphs are part of the solution, but for "or", as seen in part b, include areas from both graphs. Practicing with graphs can simplify identifying solution sets, making complex inequalities easier to understand.