Problem 55 - Algebra 2

Question

What is the solution set of the inequality \(-20 < 4 x-8 < 12 ?\) \(\mathbf{F}-7 < x < 1\) \(\mathbf{G}-3 < x < 5\) \(\mathbf{H}-7 < x < 5\) \(\mathbf{J}-3 < x < 1\)

Step-by-Step Solution

Answer

The solution set is option G: \(-3 < x < 5\).

Step 1: Understand the Inequality

We have a compound inequality: \(-20 < 4x - 8 < 12\). This means that the expression \(4x - 8\) must simultaneously be greater than -20 and less than 12.

Step 2: Solve the Left Inequality

First, solve \(-20 < 4x - 8\).Add 8 to both sides:\(-20 + 8 < 4x\)which simplifies to:\(-12 < 4x\).Now, divide by 4 to solve for \(x\):\(-3 < x\).

Step 3: Solve the Right Inequality

Next, solve \(4x - 8 < 12\).Add 8 to both sides:\(4x < 20\).Divide by 4 to solve for \(x\):\(x < 5\).

Step 4: Combine the Solutions

Combine the two parts of your solution to write the solution to the compound inequality:\(-3 < x < 5\).

Step 5: Match with Given Options

Compare the solution \(-3 < x < 5\) with the given options:- Option \(F\): \(-7 < x < 1\)- Option \(G\): \(-3 < x < 5\)- Option \(H\): \(-7 < x < 5\)- Option \(J\): \(-3 < x < 1\)The correct answer is option \(G\): \(-3 < x < 5\).

Key Concepts

Compound Inequality

A compound inequality involves multiple inequality expressions that are combined in a way to find values of a variable that satisfy all the inequalities involved. In our example exercise, we have the compound inequality \(-20 < 4x - 8 < 12\). This means that the expression \(4x - 8\) must be greater than -20 and less than 12 simultaneously. Each part of the compound inequality has to be satisfied for the entire expression to be true.There are two main types of compound inequalities:

• "And" compound inequalities: The solution set consists of values that satisfy both inequalities simultaneously, like our example.
• "Or" compound inequalities: The solution set consists of values that satisfy at least one of the inequalities.

The critical step in solving a compound inequality is to handle each inequality separately and then combine their solutions, as demonstrated in our example solution where two inequalities \(-20 < 4x - 8\) and \(4x - 8 < 12\) were solved independently.

Solution Set

The solution set of an inequality comprises all the possible values of the variable that satisfy the given condition(s). For the compound inequality \(-20 < 4x - 8 < 12\), the solution set represents all values of \(x\) that make both inequalities true.In the step-by-step solution, we found that solving the two parts of the compound inequality resulted in \(-3 < x < 5\). This means that any value of \(x\) between -3 and 5 (not including -3 and 5 themselves) satisfies the compound inequality.Here are some key points about solution sets:

• They are often represented in interval notation or as a range. For the given example, it could be written as \((-3, 5)\).
• It is essential for solution sets to be verified by checking values that are within and outside the boundaries to ensure they satisfy the original inequality.
• In compound inequalities, always consider whether the inequalities are strict (\( < \) and \( > \)) or inclusive (\( \leq \) or \( \geq \)). This affects whether the boundary values are included in the solution set.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations or inequalities to find the value of the unknown variable. In our problem, algebraic manipulation was essential to solve the compound inequality \(-20 < 4x - 8 < 12\). We broke it down into two simpler inequalities and tackled each one individually.Let's revisit the step-by-step breakdown:

• For the left inequality \(-20 < 4x - 8\), we added 8 to both sides, giving \(-12 < 4x\), and then divided by 4, resulting in \(-3 < x\).
• For the right inequality \(4x - 8 < 12\), we added 8 to get \(4x < 20\), followed by division by 4, leading to \(x < 5\).

These steps are crucial in understanding algebraic manipulation:

• Adding or subtracting the same value from both sides of the inequality does not change the inequality's direction.
• When multiplying or dividing both sides of an inequality by a positive number, the direction remains the same. However, dividing or multiplying by a negative number reverses the inequality direction.
• Always perform operations carefully to avoid mistakes that can lead to incorrect solutions.