Problem 27 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -7 \leq \frac{1-4 x}{7}<12 $$

Step-by-Step Solution

Answer

The solution set is \((-20.75, 12.5]\) in interval notation.

Step 1 - Break Down the Double Inequality

The given inequality is \(-7 \leq \frac{1-4x}{7} < 12\). This is a compound inequality, meaning we need to solve it in two parts: \(-7 \leq \frac{1-4x}{7}\) and \(\frac{1-4x}{7} < 12\). Let's handle each inequality separately.

Step 2 - Solve the Left Inequality

First, solve \(-7 \leq \frac{1-4x}{7}\).1. Multiply both sides by 7 to eliminate the fraction: \(-49 \leq 1 - 4x\).2. Subtract 1 from both sides: \(-50 \leq -4x\).3. Divide by -4, remember to reverse the inequality sign: \(\frac{50}{4} \geq x\) or \(12.5 \geq x\). So, \(x \leq 12.5\).

Step 3 - Solve the Right Inequality

Now solve \(\frac{1-4x}{7} < 12\).1. Multiply both sides by 7 to eliminate the fraction: \(1 - 4x < 84\).2. Subtract 1 from both sides: \(-4x < 83\).3. Divide by -4 and reverse the inequality sign: \(x > -\frac{83}{4}\) or \(x > -20.75\).

Step 4 - Combine Both Solutions

Combine the solutions obtained from both parts. We have \(x \leq 12.5\) from the left inequality and \(x > -20.75\) from the right inequality. The solution set can be expressed as the compound inequality: \(-20.75 < x \leq 12.5\).

Step 5 - Express the Solution in Interval Notation

Translate the combined inequality \(-20.75 < x \leq 12.5\) into interval notation. This is equivalent to the interval:\((-20.75, 12.5]\).

Key Concepts

Interval Notation

Interval notation is a shorthand way to express inequalities in mathematics. It is especially useful when dealing with solution sets of inequalities. By using interval notation, one can easily understand and communicate the range of possible values that a variable can take.

The following are some basic symbols and concepts commonly used in interval notation:

• A round bracket, "(", or ")", indicates that the endpoint is not included in the interval.
• A square bracket, "[", or "]", means the endpoint is included in the interval.

Let's consider the example from the exercise where the solution set is given by the inequality e.g., \(-20.75 < x \leq 12.5\)
In interval notation, this inequality is represented as \((-20.75, 12.5]\):

• The parentheses "(" means that \(-20.75\) is not included in the interval (i.e., it can take a value greater than \(-20.75\), but not \(-20.75\) itself).
• The square bracket "]" indicates that \(12.5\) is included in the interval (i.e., \(x\) can be equal to \(12.5\)).

Interval notation provides a concise solution set representation, simplifying the communication of complex mathematical expressions.

Compound Inequality

A compound inequality involves two separate inequalities joined by the word "and" or "or." This type of inequality is used to describe a range in which a value can exist. Compound inequalities are often seen in a form similar to the exercise example: \(-7 \leq \frac{1-4x}{7} < 12\).

When you see a compound inequality like this, treat it as two distinct inequalities combined. Here:

• \(-7 \leq \frac{1-4x}{7}\)
• \(\frac{1-4x}{7} < 12\)

The "and" conjunction means that both conditions must be satisfied by the variable. By solving each inequality separately, you ultimately combine the two solutions to form one compound answer.

In simple terms:

• The first inequality gives one side of the interval.
• The second inequality gives the other side of the interval.

Joining these solutions creates a complete understanding of where the variable can lie within a particular range.

Solution Set

In solving inequalities, the solution set is the range of all possible values that satisfy the inequality. It's a fundamental concept, as solving for these values allows us to find where a mathematical statement holds true.

In the exercise, the inequality\(-20.75 < x \leq 12.5\) is the solution
By determining the solution set, you effectively understand the possible values \(x\) can take which satisfy both pieces of the compound inequality.The process involves:

• Solving each part of the compound inequality individually, such as \(-7 \leq \frac{1-4x}{7}\) and \(\frac{1-4x}{7} < 12\).
• Combining the results to reflect the entire range of solutions.

Expressing the solution in interval notation \((-20.75, 12.5]\) further clarifies the solution set by concisely displaying the boundaries within which \(x\) must exist. Understanding the solution set connects the mathematical process to real-world applications, allowing for practical analysis and decision-making. This clarity is key for correctly interpreting and applying mathematical concepts.