Problem 17 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ 3 x-5>2 x-8 $$

Step-by-Step Solution

Answer

Solution set: \( \{ x | x > -3 \} \).

Step 1: Move Variable Terms to One Side

Start by subtracting \(2x\) from both sides of the inequality to bring all variable terms to one side. This gives us: \(3x - 2x - 5 > -8\). Simplified, it becomes: \(x - 5 > -8\).

Step 2: Isolate the Variable

Next, add 5 to both sides of the inequality to isolate \(x\). This yields: \(x - 5 + 5 > -8 + 5\), which simplifies to: \(x > -3\).

Step 3: Write Solution in Set Notation

Express the solution \(x > -3\) in solution set notation: \( \{ x | x > -3 \} \).

Step 4: Graph the Solution Set

To graph the inequality \(x > -3\), draw a number line. Mark \(-3\) with an open circle to indicate that it is not included in the solution set, and shade the line to the right of \(-3\) to represent all numbers greater than \(-3\).

Key Concepts

Solution Set Notation

When solving algebraic inequalities, representing the solution as a set of numbers is crucial. Solution set notation is a concise way to express all possible solutions that satisfy the inequality. For instance, if we have the inequality \(x > -3\), its solutions are all numbers greater than \(-3\).

• The notation \(\{ x | x > -3 \}\) is read as "the set of all \(x\) such that \(x\) is greater than \(-3\)."
• The vertical line \(|\) stands for "such that" and helps define the condition of the solutions.
• Braces \(\{ \}\) are used to encapsulate the set, indicating that it is a collection of numbers.

This notation is a powerful tool in algebra, especially when dealing with inequalities, because it provides clarity and precision. When preparing solution sets, ensure that you accurately reflect the range or specific conditions identified by your inequality.

Graphing Inequalities

Graphing inequalities on a number line is an effective way to visualize the set of solutions. It helps to see not only the values that satisfy an inequality but also the transition points, like critical points where the inequality switches from true to false.

• To graph \(x > -3\), start by drawing a straight horizontal line to serve as the number line.
• Identify and mark the critical point, \(-3\), with an open circle. This signifies that \(-3\) is not included in the solution set.
• Shade the portion of the line to the right of \(-3\) to indicate all numbers greater than \(-3\). This shaded area shows the solutions for the inequality.

By using graphing, you can quickly determine whether particular values satisfy the inequality or not. It's also an excellent method for comparing or combining multiple inequalities.

Solving Inequalities Step-by-Step

Solving inequalities involves steps similar to solving equations, with some additional considerations. It's crucial to manipulate the inequality carefully to ensure the solution remains valid. Let's go through the steps necessary to solve the inequality \(3x - 5 > 2x - 8\). **Step 1: Move Variable Terms to One Side** Start by simplifying the inequality. Subtract \(2x\) from both sides to get all variable terms on one side, resulting in \(x - 5 > -8\). This simplifies the inequality.**Step 2: Isolate the Variable** Next, isolate \(x\) by adding \(5\) to both sides of the inequality: \(x > -3\). This step ensures \(x\) stands alone on one side of the equation, making the inequality easier to interpret.**Key Considerations**- When multiplying or dividing both sides by a negative number, the direction of the inequality must be reversed. This rule ensures the inequality remains true.- Always simplify completely in each step to avoid errors.By following these structured steps, solving inequalities becomes a clearer process, helping you avoid pitfalls and guaranteeing accurate solutions.