Problem 18 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 0<5-2 x $$

Step-by-Step Solution

Answer

The solution is \((-\infty, \frac{5}{2})\).

Step 1: Rearrange the Inequality

Start with the given inequality: \[0 < 5 - 2x\]Rearrange this by moving all terms involving the variable \(x\) to one side of the inequality. Subtract 5 from both sides to isolate the term with \(x\):\[0 - 5 < -2x\]Which simplifies to:\[-5 < -2x\]

Step 2: Solve for x

Now, solve for \(x\) by dividing both sides by \(-2\). Remember, when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign:\[\frac{-5}{-2} > x\]Simplifying gives:\[\frac{5}{2} > x\]Or equivalently, \(x < \frac{5}{2}\).

Step 3: Express the Solution in Interval Notation

Since \(x\) should be less than \(\frac{5}{2}\), the interval notation would be:\[(-\infty, \frac{5}{2})\]

Step 4: Graph the Solution Set

To graph the solution on a number line:1. Draw a number line.2. Mark \(\frac{5}{2}\) with an open circle, indicating that \(\frac{5}{2}\) is not included in the solution set.3. Shade the region to the left of \(\frac{5}{2}\) to represent all values less than \(\frac{5}{2}\).

Key Concepts

Interval Notation

Interval notation is a mathematical way to express all the numbers between two endpoints. It's especially useful in showing solutions for inequalities because it clearly indicates the range of values that satisfy the inequality.
For a linear inequality like the one we solved, we ended up with "\(-\infty, \frac{5}{2}\)". The parentheses "(" and ")" show that these endpoints are not included.
When you see \(-\infty\), it indicates that the number range goes on indefinitely to negative numbers. It's crucial since linear inequalities often have solutions extending in one direction.

• An open circle on a graph corresponds to a parenthesis in interval notation, showing that the endpoint is not part of the solution.
• If the endpoint were inclusive, we'd use square brackets "[" or "]".

Understanding interval notation makes it easier to communicate the range of numbers that satisfy the inequality.

Graphing Inequalities

Graphing inequalities is about visually showing which numbers satisfy an inequality on a number line. By graphing, you can easily see which numbers make the inequality true.
For the inequality \(x < \frac{5}{2}\), the steps involve:

• Drawing a number line.
• Marking the number \(\frac{5}{2}\) with an open circle. This shows that this specific point is not included in the solution. This visually represents the inequality condition clearly.
• Shading the area to the left of \(\frac{5}{2}\) to represent all numbers less than \(\frac{5}{2}\). The shaded region helps in quickly identifying the numbers that satisfy the inequality condition.

Graphing inequalities helps in understanding how the range of numbers changes with the inequality and makes it easy to interpret the solution.

Solution Set

The solution set of an inequality is the collection of all values that make the inequality true. For our linear inequality exercise, the solution set represents every possible value of\(x\) that satisfies \(x < \frac{5}{2}\).
The solution set, in this case, is every number less than \(\frac{5}{2}\).
Expressed in interval notation, it was \((-\infty, \frac{5}{2})\).

• This notation excludes \(\frac{5}{2}\) itself, proving it by the open interval point.
• The "-\infty" further extends the solution set infinitely in the negative direction.

Understanding the solution set conceptually is vital because it tells you exactly which numbers work in your inequality. Knowing how to express a solution set ensures clarity and consistency, particularly when switching between symbolic, algebraic, and graphical representations of inequalities.