Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 3 x-2>14 $$

Step-by-Step Solution

Answer

The solution is \((\frac{16}{3}, \infty)\).

Step 1: Isolate the Variable Expression

To solve the inequality, we need to isolate the variable expression on one side of the inequality. Start by eliminating the constant term on the left. Add 2 to both sides of the inequality:\[ 3x - 2 + 2 > 14 + 2 \]This simplifies to:\[ 3x > 16 \]

Step 2: Solve for the Variable

Next, divide both sides by 3 to solve for \(x\):\[ \frac{3x}{3} > \frac{16}{3} \]This simplifies to:\[ x > \frac{16}{3} \]

Step 3: Express the Solution in Interval Notation

The solution \( x > \frac{16}{3} \) describes all values greater than \(\frac{16}{3}\). In interval notation, this is written as:\( \left( \frac{16}{3}, \infty \right) \).

Key Concepts

Solving Inequalities

Solving inequalities is quite similar to solving regular equations. However, when solving inequalities, the solution is not just one number. Instead, it's a range of numbers that satisfy the inequality condition. Here's how you can approach solving them:

• First, treat the inequality sign just like an equal sign.
• Perform the same operations you would do to solve an equation such as addition, subtraction, multiplication, or division on both sides.
• The goal is to isolate the variable on one side to understand what values satisfy the inequality.

An important rule to remember is when you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. This is a special rule that distinguishes inequalities from equations.

Interval Notation

Interval notation is a system used to express a set of numbers, typically as a solution to an inequality. It's a very efficient way of showing continuous ranges of numbers, as opposed to listing them all. Let's explore how it works:

• Use round brackets, \( ( \) or \( ) \), for values that are not included in the set, termed as "open interval".
• Use square brackets, \( [ \) or \( ] \), when the endpoint value is included, known as "closed interval".
• An interval from \( a \) to \( b \) (not including \( a \) or \( b \)) will look like \( (a, b) \).
• If the interval starts or goes to infinity, \( \infty \) or \( -\infty \) is always in round brackets, since infinity is not a number we can reach or include.

For the inequality solution \( x > \frac{16}{3} \), we would use interval notation \( \left( \frac{16}{3}, \infty \right) \) to show all numbers greater than \( \frac{16}{3} \).

Inequality Solutions

Inequality solutions represent a set of numbers that make an inequality true. They can be expressed in different ways, including expressions, graphs, or interval notation:

• In expressions, the solution is shown with inequalities, like \( x > \frac{16}{3} \), indicating any number greater than \( \frac{16}{3} \) is a solution.
• Graphically, inequalities can be represented on a number line where shaded regions represent solutions, with open or closed dots indicating whether endpoints are included.
• Interval notation is another concise way to document solutions, helpful in both visualizing and writing solutions succinctly.

In our example, the inequality \( 3x - 2 > 14 \) simplifies to \( x > \frac{16}{3} \). Here, the meaningful solution represents all \( x \)-values that are greater than \( \frac{16}{3} \). To check, substitute some numbers larger than \( \frac{16}{3} \) back into the original inequality to see if it holds true.