Problem 27 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -15 x+34<-15 $$

Step-by-Step Solution

Answer

\(x > \frac{49}{15}\); interval: \(\left( \frac{49}{15}, \infty \right)\).

Step 1: Simplify the Inequality

Start by simplifying the inequality \[ -15x + 34 < -15 \]by subtracting 34 from both sides to isolate terms involving \(x\):\[ -15x + 34 - 34 < -15 - 34 \]which simplifies to:\[ -15x < -49 \]

Step 2: Solve for x

To solve for \(x\), divide both sides of the inequality by -15. Remember that dividing or multiplying by a negative number reverses the inequality sign:\[ x > \frac{49}{15} \]

Step 3: Graph the Solution Set

On a number line, plot an open dot at \(\frac{49}{15}\) (which is approximately 3.27) to indicate that this point is not included, and shade the region to the right to represent all numbers greater than \(\frac{49}{15}\).

Step 4: Write in Interval Notation

The solution set in interval notation is represented as:\[ \left( \frac{49}{15}, \infty \right) \]

Key Concepts

Inequality Graphing

Graphing an inequality can help you visually understand the solution set. The inequality we solved was \( x > \frac{49}{15} \). To graph this on a number line, start by drawing a number line with values marked. Determine where \( \frac{49}{15} \) lies; since it is approximately 3.27, place it accordingly on the number line.Use an open dot at \( \frac{49}{15} \) because the inequality is a strict inequality (greater than but not equal to). This dot indicates that \( \frac{49}{15} \) itself is not part of the solution.Shade the line to the right of the open dot. This shading extends to the right indefinitely, illustrating that all numbers greater than \( \frac{49}{15} \) are solutions to the inequality. Graphing inequalities like this provides a clear picture of where solutions lie on a number line.

Interval Notation

Interval notation is a way of describing sets of numbers along a number line, focusing especially on solutions to inequalities. For our inequality \( x > \frac{49}{15} \), the solution set does not include \( \frac{49}{15} \) itself, only numbers greater than it.To express this, we use a parenthesis to signify that \( \frac{49}{15} \) is not included. So, the interval begins with \( \left( \frac{49}{15} \right) \).The set of all numbers greater than a particular number keeps going forever, which is represented by \( \infty \) in interval notation. Hence, we write the solution as \( \left( \frac{49}{15}, \infty \right) \). Parentheses are used again around \( \infty \), as infinity is a concept, not a number that can be "reached."

Algebraic Simplification

Solving inequalities often involves simplifying algebraic expressions to make finding a solution easier. The initial inequality was \( -15x + 34 < -15 \). The first step in simplification is to isolate the term involving \( x \). You achieve this by subtracting 34 from both sides. This step reduces the inequality to \( -15x < -49 \).Next, you need to solve for \( x \) by dividing both sides by -15. A critical point here is remembering that dividing or multiplying an inequality by a negative number reverses the inequality sign. Thus, -15 divided by -15 reverses \( < \) to \( > \). So the expression becomes \( x > \frac{49}{15} \).The main takeaway from algebraic simplification is to perform operations carefully while following rules unique to inequalities, like reversing the signs when necessary.