Problem 97 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$-\frac{1}{2} x+0.7 x-5>0$$

Step-by-Step Solution

Answer

The solution set is \((25, \infty)\).

Step 1: Combine Like Terms

Given the inequality \[-\frac{1}{2}x + 0.7x - 5 > 0\]we start by combining the terms with \(x\). The terms \(-\frac{1}{2}x\) and \(0.7x\) can be combined by converting \(0.7\) into a fraction which is \(\frac{7}{10}\). Now let's combine the coefficients:\[-\frac{1}{2} + \frac{7}{10} = -\frac{10}{20} + \frac{14}{20} = \frac{4}{20} = \frac{1}{5}\].So the inequality becomes:\[\frac{1}{5}x - 5 > 0\]

Step 2: Isolate the Variable Term

Next, we'll isolate the \(x\) term. Add 5 to both sides of the inequality to get the variable term by itself on one side:\[\frac{1}{5}x > 5\]

Step 3: Solve for the Variable

To solve for \(x\), multiply both sides of the inequality by 5, which is the reciprocal of \(\frac{1}{5}\). This step gets \(x\) by itself:\[x > 5 \times 5\]Simplifying gives:\[x > 25\]

Step 4: Write the Solution in Interval Notation

The solution \(x > 25\) can be expressed in interval notation as:\[(25, \infty)\]

Step 5: Graph the Solution on a Number Line

On a number line, \((25, \infty)\) will be represented by an open circle at 25 and a line extending to the right to indicate that all numbers greater than 25 are included in the solution set.

Key Concepts

Interval Notation

Interval notation is a way to express a range of values compactly. It uses brackets and parentheses to indicate which endpoints are included or excluded. For instance, in our example, the inequality \(x > 25\) is translated into interval notation as \((25, \infty)\).

This tells us that the solution includes all numbers greater than 25, but not 25 itself. The parenthesis around 25 signifies that 25 is not part of the solution set, while infinity always carries a parenthesis since it is not a real number that can be reached or included.

The notation is a clean and efficient way to communicate mathematical solutions involving continuous sets of numbers. It is especially useful in calculus and algebra where precision in expressing ranges is crucial.

Graphical Representation

Graphical representation is an intuitive way to visualize mathematical concepts, including inequalities. Solving the inequality \(x > 25\) can be easily depicted on a number line.

Here’s how it’s done:

• First, identify 25 on the number line. Place an open circle (or hollow dot) at this point to signify that 25 is not included in the solution.
• Next, draw a line or an arrow extending to the right from 25, indicating all numbers greater than 25 are solutions.

The representation provides a quick visual snapshot of the solution set, making it accessible even at a glance. This approach helps clarify which values satisfy the inequality, aiding in understanding and communication of mathematical ideas.

Algebraic Manipulation

Algebraic manipulation is the process of restructuring equations or inequalities to simplify or solve them. It's a crucial skill in math that allows us to isolate variables and find their values.

In our problem, we performed several essential manipulations:

• First, we combined like terms: \(-\frac{1}{2}x + 0.7x\). Converting 0.7 to a fraction \(\frac{7}{10}\), we found the combined coefficient to be \(\frac{1}{5}x\).
• Next, we isolated the \(x\) term by adding 5 to both sides, leading to \(\frac{1}{5}x > 5\).
• Finally, we multiplied both sides by 5 to solve for \(x\), giving us \(x > 25\).

Algebraic manipulation allows solving even complex equations more efficiently, and gaining proficiency in these techniques is essential for tackling higher-level math problems.

Solving Inequalities

Solving inequalities involves finding all possible values of variables that make the inequality true. Unlike equations that have single or several discrete solutions, inequalities often involve entire ranges.

For instance, to solve \(-\frac{1}{2}x + 0.7x - 5 > 0\), we combined the variable terms, simplified the expression, and performed algebraic operations to isolate \(x\). The solution \(x > 25\) was reached by multiple steps:

• Combine like terms, turning the inequality into \(\frac{1}{5}x - 5 > 0\).
• Isolate the \(x\) term with addition: \(\frac{1}{5}x > 5\).
• Eliminate the fraction by multiplying, yielding \(x > 25\).

The set of all numbers satisfying the inequality is expressed in interval notation as \((25, \infty)\).
Understanding the solving process equips students with the tools to tackle similar inequalities effectively.