Problem 14 - Calculus: One and Several Variables

Question

Solve the inequality. Express the solution as an interval or as the union of intervals. Mark the solution on a number line. $$3 x+5<\frac{1}{2}(4-x)$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > -\frac{6}{7}\). So the set of all solutions in interval notation is \((- \frac{6}{7}, \infty)\).

Step 1: Simplify the equation

Start by simplifying the inequality. Distribute \(\frac{1}{2}\) on the right side to simplify it. So, \(3x + 5 < 2 - \frac{1}{2}x\).

Step 2: Re-arrange terms

Next, put all terms involving \(x\) on one side and the constants on the other. To do this, add \(\frac{1}{2}x\) to both sides and subtract 5 from both sides. This leads to \(3x + \frac{1}{2}x > 2 -5.\)

Step 3: Simplify

Simplify the inequality to make it easier to solve. This gives \(3.5 x > -3.\)

Step 4: Solve for x

Now, solve the inequality for \(x\). Do this by dividing both sides by 3.5. This provides the solution \(x > -\frac{6}{7}\).

Key Concepts

Algebraic Inequalities

When we talk about algebraic inequalities, we're referring to mathematical statements that express the relative order between two different algebraic expressions. Inequalities use symbols like <>, \textless, \textgreater, \textless=, and \textgreater= to show that one side is less than, greater than, less than or equal to, or greater than or equal to the other side. Solving inequalities involves finding all the possible values of the variable that make the inequality true.

For example, let's take the inequality from the exercise: \(3x + 5 < \frac{1}{2}(4 - x)\). The goal is to isolate the variable \(x\) on one side of the inequality to find the range of values that satisfies the condition. The process is similar to solving an equation: you perform operations to simplify and rearrange the inequality. Just remember that if you multiply or divide by a negative number, the inequality symbol must be flipped to maintain the correct relation. With algebraic inequalities, it's important to keep number sense in mind and remember that you're dealing with a range of values, not just a single answer.

Interval Notation

In mathematics, interval notation is a way of representing a set of numbers on the real number line. It gives us a concise method to write down ranges of values that satisfy an inequality. The notation uses parentheses and brackets to describe intervals. Parentheses, ( or ), indicate that the endpoint is not included in the interval—these are called open intervals. Brackets, [ or ], mean that the endpoint is included—these are closed intervals.

Using the inequality \(x > -\frac{6}{7}\), we can express the solution in interval notation. Since \(x\) is greater than \(-\frac{6}{7}\), but \(-\frac{6}{7}\) is not included in the solution set, we use a parenthesis. We also know that \(x\) can be any number greater than \(-\frac{6}{7}\) up to infinity, which we symbolize with the infinity symbol (∞). Thus, the interval notation for our solution is \((-\frac{6}{7}, ∞)\).

Number Line Representation

The number line representation is a visual tool that helps to better understand inequalities and intervals. A number line is simply a straight line with numbers placed at equal intervals along its length. In the case of the inequality \(x > -\frac{6}{7}\), we show the solution on a number line by drawing a ray starting at \(-\frac{6}{7}\) and extending to the right, towards positive infinity.

On the number line, we would represent the endpoint \(-\frac{6}{7}\) with an open circle, since that value is not included in the solution set (as indicated by the greater than symbol). Then, we would shade or draw an arrow to the right of \(-\frac{6}{7}\) to show all the values of \(x\) that are part of the solution interval. The number line effectively gives us a clear, at-a-glance depiction of the solution to the inequality.