Problem 30 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -7(x-2)+1<15 $$

Step-by-Step Solution

Answer

The solution is \\(x > 0\\), or in interval notation, \\((0, \infty)\\).

Step 1: Distribute the Negative Number

To eliminate the parentheses, we need to distribute \(-7\) across the terms inside the parentheses. This gives us \(-7 \times x + (-7) \times (-2)\). Simplifying this step, we have: \(-7x + 14\).

Step 2: Combine and Simplify the Inequality

Substitute the distributed terms back into the inequality: \(-7x + 14 + 1 < 15\). Then, combine like terms: \(-7x + 15 < 15\).

Step 3: Isolate the Variable Term

Subtract 15 from both sides to begin isolating the variable \(x\): \(-7x + 15 - 15 < 15 - 15\), which simplifies to: \(-7x < 0\).

Step 4: Solve for the Variable

To solve for \(x\), divide both sides by \(-7\). Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. Thus, dividing by \(-7\) gives \(x > 0\).

Step 5: Express the Solution in Interval Notation

In interval notation, the solution \(x > 0\) is expressed as \((0, \infty)\).

Step 6: Graph the Solution Set

On a number line, draw an open circle at \(x = 0\) to represent that \(0\) is not included in the solution set. Shade the line to the right of \(x = 0\) to indicate that the solution includes all numbers greater than \(0\).

Key Concepts

Distributive Property

The distributive property is a useful tool when dealing with equations and inequalities involving parentheses. It allows us to multiply a single term by each term inside a set of parentheses. In our exercise, we started with

• \(-7(x-2)+1<15\)

To eliminate the parentheses, we distributed \(-7\) across \((x-2)\). This means we multiplied \(-7\) by both \(x\) and \(-2\), giving us

• \(-7x + 14\)

The key to the distributive property is ensuring that each term inside the parentheses is accounted for in the multiplication. Once the distribution is complete, we can proceed with solving the inequality, simplifying it in each step. This step-by-step approach makes solving complex inequalities much simpler and more manageable.

Interval Notation

Interval notation provides a concise way to express ranges of numbers. It's particularly useful for expressing the solution sets of inequalities. After solving

• \(-7(x-2)+1<15\)

We found the solution to be \(x > 0\). In interval notation, this is represented as

• \((0, \infty)\)

Here's what the symbols mean:

• \((0, \infty)\): The round bracket "(" indicates that 0 is not included in the solution set, hence the open interval. "\(\infty\)" signifies that there is no upper bound to the solution, extending forever to the right.

Interval notation is advantageous as it clearly shows when endpoints are included (closed intervals) or not (open intervals) in the solution set.

Graphing on a Number Line

Graphing inequalities on a number line is an excellent way to visually represent solution sets. For our inequality \(x > 0\), here's how to graph it:

• Draw a number line and mark the point \(x = 0\).
• Place an open circle at \(x = 0\) to show that 0 is not included in the solution.
• Shade the line to the right of 0 to indicate that all numbers greater than 0 are included in the solution.

The open circle is a simple yet powerful visual indicator that differentiates values not included in a solution from those that are. This method provides a clear, intuitive representation of the inequality's solution. Graphing on a number line helps in understanding the range and scope of possible solutions, making it easier to visualize and confirm the solution set.