Problem 11 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 x-5>3 $$

Step-by-Step Solution

Answer

The solution is \( x > 4 \) or \( (4, \infty) \).

Step 1: Starting Inequality

We start with the inequality:\[ 2x - 5 > 3 \]Our goal is to isolate \( x \) by first focusing on simplifying the inequality.

Step 2: Add 5 to Both Sides

To isolate \( x \), we first eliminate the \( -5 \) by adding 5 to both sides of the inequality:\[ 2x - 5 + 5 > 3 + 5 \]This simplifies to:\[ 2x > 8 \]

Step 3: Divide Both Sides by 2

Next, divide each side of the inequality by 2 to solve for \( x \):\[ \frac{2x}{2} > \frac{8}{2} \]This reduces to:\[ x > 4 \]

Step 4: Express Solution in Interval Notation

The solution to the inequality \( x > 4 \) can be expressed in interval notation as:\( (4, \infty) \)

Step 5: Graph the Solution

On a number line, indicate the solution \( x > 4 \) by drawing an open circle at \( 4 \) (indicating 4 is not included in the solution set), and shade all numbers greater than 4 extending indefinitely to the right.

Key Concepts

Solving Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, often using symbols like ">", "<", "≥", and "≤".
Solving inequalities involves finding the range of values that satisfy the inequality condition. The process is similar to solving equations, but with a crucial distinction: when multiplying or dividing both sides by a negative number, the inequality sign flips.
Let's solve the inequality:

• Start with the inequality: \(2x - 5 > 3\)
• Add 5 to each side to simplify: \(2x > 8\)
• Divide both sides by 2 to solve for \(x\): \(x > 4\)

This result, \(x > 4\), tells us that any number greater than 4 satisfies the inequality. Remember to apply the rule of flipping the inequality sign only if you multiply or divide both sides by a negative number.

Interval Notation

Interval notation is a compact way to express the set of solutions for inequalities. It uses brackets and parentheses to show whether endpoints are included or excluded:

• \([a, b]\) means all numbers between \(a\) and \(b\), inclusive of both endpoints.
• \((a, b)\) means all numbers between \(a\) and \(b\), excluding both endpoints.
• \([a, b)\) or \((a, b]\) indicates inclusion of one endpoint and exclusion of the other.
• An infinity symbol \((\infty)\) is used when there is no bound on one side. Infinity is always accompanied by a parenthesis, never a bracket, because infinity is not an actual number we can reach.

For the solution \(x > 4\), we use interval notation to express this as \((4, \infty)\). The parenthesis around 4 indicates that it is not included in the solution set.

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set. This method emphasizes what values of \(x\) are included or excluded in the inequality:
To graph \(x > 4\) on a number line:

• Draw a number line and locate the number 4.
• Place an open circle (instead of a filled dot) at 4 because 4 is not part of the solution.
• Shade the line to the right of 4, extending infinitely, to represent all numbers greater than 4.

This visual representation complements the mathematical expression, depicting clearly the range of possible values that fulfill the inequality \(x > 4\). Remember, an open circle means the point itself is not included in the solution, whereas a closed dot would indicate inclusion.