Problem 16 - Algebra and Trigonometry

Question

\(9-32\) me 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: Isolate the variable term

First, we isolate the term containing the variable. Begin by subtracting 5 from both sides of the inequality:\[0 < 5 - 2x \quad \Rightarrow \quad 0 - 5 < -2x \quad \Rightarrow \quad -5 < -2x\]

Step 2: Solve for the variable

Now, solve for \(x\) by dividing both sides of the inequality by \(-2\). Remember to flip the inequality sign because we are dividing by a negative number:\[\frac{-5}{-2} > x \quad \Rightarrow \quad \frac{5}{2} > x\]This can also be expressed as \(x < \frac{5}{2}\).

Step 3: Express in interval notation

The interval notation for \(x < \frac{5}{2}\) is:\[(-\infty, \frac{5}{2})\]

Step 4: Graph the solution set

To graph \(x < \frac{5}{2}\), draw a number line and shade all numbers to the left of \(\frac{5}{2}\). Put an open circle at \(\frac{5}{2}\) to indicate that \(\frac{5}{2}\) is not included in the solution set.

Key Concepts

Interval Notation

Interval notation is a mathematical way to describe the set of solutions you find when solving inequalities. It uses brackets and parentheses to show the range of numbers included or excluded in the solution set.
For example, when you have an inequality like \(x < \frac{5}{2}\), you express it in interval notation as \((-\infty, \frac{5}{2})\).

• Parentheses ( ): Used to indicate that a number is not included in the interval. So, \((-\infty, \frac{5}{2})\) means every number up to \(\frac{5}{2}\) is included, but \(\frac{5}{2}\) itself is not.
• Brackets [ ]: If a number is included, you use a bracket. For example, \([a, b]\) means that both \(a\) and \(b\) are included in the interval.
• Infinity (and negative infinity): Always paired with a parenthesis, since they aren’t actual numbers you can reach. Use \(-\infty\) or \(\infty\) to indicate that the interval extends indefinitely in either direction.

Interval notation offers a compact, clear way to represent solution sets and is frequently used in high school and college mathematics.

Inequality Solving

Solving inequalities is similar to solving equations, but with a few important differences. The inequality sign shows the relationship between expressions. In the exercise given, you have the inequality \(0 < 5 - 2x\).

Here's a simple process to follow:

• Step 1 - Isolate the variable: Get the term with the variable by itself. For the example, subtract 5 from each side to get \(-5 < -2x\).
• Step 2 - Solve the inequality: Divide or multiply to solve for the variable like an equation, but always remember: when you multiply or divide both sides by a negative, you must flip the inequality sign! So, \(-5 < -2x\) becomes \(\frac{5}{2} > x\).

Understanding these steps will improve your skills in handling inequalities and help you avoid common mistakes, like forgetting to flip the inequality sign.

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set of an inequality. In this particular example, we need to graph \(x < \frac{5}{2}\). Here’s a simple guide:

• Draw a number line: Start by drawing a horizontal line and marking zero and key numbers, like \(\frac{5}{2}\).
• Use circles to indicate inclusion/exclusion: Place an open circle at \(\frac{5}{2}\) because it’s not included in the solution (the inequality is \(<\), not \(\leq\)). For inequalities like \(\leq\) or \(\geq\), use a closed circle.
• Shade the appropriate side: Since it's \(x < \frac{5}{2}\), shade all numbers to the left of \(\frac{5}{2}\). This indicates all numbers less than \(\frac{5}{2}\) are part of the solution set.

Graphing inequalities is a powerful tool that makes it easier to see and understand the range of possible solutions, enhancing your grasp on both solving and representing inequalities visually.