Problem 36 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -\frac{3}{4}<\frac{2-t}{5}<\frac{3}{4} $$

Step-by-Step Solution

Answer

\( (-\frac{7}{4}, \frac{23}{4}) \)

Step 1: Understand the inequality structure

The given compound inequality is \(-\frac{3}{4} < \frac{2-t}{5} < \frac{3}{4}\). It is a double inequality, meaning we need to solve two inequalities: \(-\frac{3}{4} < \frac{2-t}{5}\) and \(\frac{2-t}{5} < \frac{3}{4}\).

Step 2: Solve the first inequality

For the inequality \(-\frac{3}{4} < \frac{2-t}{5}\), multiply all sides by 5 to clear the fraction, obtaining \(-\frac{15}{4} < 2 - t\). Add \(t\) to both sides and subtract \(-\frac{15}{4}\) from 2 to isolate \(t\), resulting in \(t < 2 + \frac{15}{4}\).

Step 3: Simplify the result of the first inequality

Calculate \(2 + \frac{15}{4}\) by converting 2 to a fraction \(\frac{8}{4}\). So, \(t < \frac{8}{4} + \frac{15}{4} = \frac{23}{4}\). This gives one part of our solution: \(t < \frac{23}{4}\).

Step 4: Solve the second inequality

For the inequality \(\frac{2-t}{5} < \frac{3}{4}\), multiply all sides by 5 again to get \(2 - t < \frac{15}{4}\). Subtract 2 from both sides to isolate \( -t\), resulting in \-t < \frac{15}{4} - 2\. Then multiply by -1 and reverse the inequality to get \(t > 2 - \frac{15}{4}\).

Step 5: Simplify the result of the second inequality

Calculate \(2 - \frac{15}{4}\) by converting 2 to a fraction \(\frac{8}{4}\). Thus, we have \(t > \frac{8}{4} - \frac{15}{4} = -\frac{7}{4}\). This gives the second part of our solution: \(t > -\frac{7}{4}\).

Step 6: Combine the solutions from both inequalities

Now combine the results from the two inequalities to form the solution set: \(-\frac{7}{4} < t < \frac{23}{4}\).

Step 7: Express the solution in interval notation

The solution \(-\frac{7}{4} < t < \frac{23}{4}\) can be expressed in interval notation as \((-\frac{7}{4}, \frac{23}{4})\), signifying that \(t\) lies within these bounds but does not include the endpoints.

Key Concepts

Compound Inequalities

Compound inequalities are mathematical statements combining two simple inequalities. In the exercise provided, the compound inequality is:\[ -\frac{3}{4} < \frac{2-t}{5} < \frac{3}{4} \]This inequality implies that the expression \(\frac{2-t}{5}\) should fall between \(-\frac{3}{4}\) and \(\frac{3}{4}\). To solve compound inequalities, both component inequalities must be solved separately. The solutions of each part need to be combined to find the full range of values satisfying the original compound inequality. It's helpful to think of compound inequalities as representing a range or section on a number line, where the solution must fit both criteria imposed by the inequalities, similar to when you need to pass two tests to qualify for a competition.

Interval Notation

Interval notation offers a concise way to express the solution sets of inequalities. For the resulting inequalities \(-\frac{7}{4} < t < \frac{23}{4}\), instead of listing several possibilities for the values of \(t\), we use interval notation:\( (-\frac{7}{4}, \frac{23}{4}) \)This notation indicates that \(t\) can take any value between \(-\frac{7}{4}\) and \(\frac{23}{4}\), but not including the endpoints. The parentheses \((, )\) are used to signify open intervals, meaning these boundary values are excluded. If the endpoints were to be included, we would use square brackets \([, ]\). With this notation, complex solution sets from compound inequalities become much easier and simpler to understand.

Symbolic Solution

Finding symbolic solutions involves expressing the solution of an inequality or equation using symbols rather than specific numbers. In the case of our inequality:\[ -\frac{3}{4} < \frac{2-t}{5} < \frac{3}{4} \]Our goal is to symbolically manipulate the inequality to find the range of \(t\). This process may involve steps such as clearing fractions by multiplying through by the denominator and performing operations on both sides to isolate \(t\). After working through the algebraic steps, the symbolic form of the solution to this inequality was derived as:\[ -\frac{7}{4} < t < \frac{23}{4} \]Using symbolic solutions allows us to handle more complex formulas without resorting immediately to numeric computation, keeping the focus on how expressions relate.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). In this exercise, the expression \(\frac{2-t}{5}\) involves subtraction and division.- **Variables in Expressions:** Here, \(t\) represents an unknown value we need to find.- **Operations:** The fraction arises as an operation applied to simplify the expression with \(t\).Dealing with algebraic expressions often requires performing operations on both sides of an inequality or equation to make it easier to solve. For example, multiplying through by 5 removed the faction in our exercise, allowing for more straightforward manipulation to isolate \(t\). Solving inequalities calls on these fundamental transformations to present solutions in a form that reflects the range of possible values the variable can take on.