Problem 72 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -5<2(x-1)-3(x+2)<5 $$

Step-by-Step Solution

Answer

The solution set is \((-13, -3)\) and can be graphed with open circles on a number line.

Step 1: Expand and Simplify the Inequality

We start by expanding both sides of the compound inequality:\[-5 < 2(x-1) - 3(x+2) < 5\]Distribute the multiplication across the terms:\[-5 < (2x - 2) - (3x + 6) < 5\]Simplify by combining like terms:\[-5 < 2x - 2 - 3x - 6 < 5\]This leads to:\[-5 < -x - 8 < 5\].

Step 2: Isolate the Variable

Now, isolate the variable \(x\) in the middle by first adding 8 to each part of the inequality:\[-5 + 8 < -x - 8 + 8 < 5 + 8\]This simplifies to:\[3 < -x < 13\].

Step 3: Solve for Positive x

To solve for \(x\), multiply all parts of the inequality by \(-1\). Remember to reverse the inequality signs:\[-3 > x > -13\].This can be rewritten as:\[-13 < x < -3\].

Step 4: Write the Solution in Interval Notation

The solution \(-13 < x < -3\) in interval notation is:\((-13, -3)\).

Step 5: Graph the Solution on a Number Line

To graph \((-13, -3)\), draw a number line and indicate open circles at \(-13\) and \(-3\), with a line connecting them, indicating all numbers in between are part of the solution. No point at either \(-13\) or \(-3\), as these are not included in the solution set.

Key Concepts

Compound Inequality

A compound inequality involves two separate inequalities joined by either "and" or "or". In our example, the compound inequality is \[-5 < 2(x-1) - 3(x+2) < 5\] This combines two inequalities in one statement. The solution must satisfy both individual inequalities simultaneously. In other words, a number is a solution only if it works for both inequalities at the same time. This is different from a simple inequality, which contains only one comparison to deal with.

• "And" implies overlap - the solutions are in the region where both inequalities are true.
• "Or" implies union - solutions are valid if they satisfy at least one of the inequalities.

In this problem, we are addressing a type of "and" compound inequality, as the variable must satisfy both parts to remain in the range of valid values.

Interval Notation

Interval notation is a concise way to express the solution to an inequality. In this case, after solving the inequality, we got: \[-13 < x < -3\] This is expressed as \((-13, -3)\) in interval notation. The round brackets indicate that -13 and -3 are not included in the solution (known as open intervals).

• Round brackets \(( )\) signify numbers that are not part of the solution.
• Square brackets \([ ]\) mean a number is included in the solution.

Interval notation is helpful because it's clear, simple, and avoids the need for inequality symbols when writing the range of solutions.

Number Line Graphing

Graphing solutions on a number line is a visual way to represent the range of numbers that satisfy an inequality. For \((-13, -3)\), we depict this interval with open circles at -13 and -3, signaling these points are not included in the solution. Draw a line connecting these points to show that all numbers between -13 and -3 are part of the solution set. This graphical representation helps visualize the extent and nature of solutions:

• Open circles indicate that endpoints are not included in the solution.
• A connected line shows all numbers within the interval are valid solutions.

Using a number line can simplify understanding complex inequalities, especially when combined with interval notation.

Variable Isolation

Isolating the variable is a critical step in solving inequalities. The goal is to have the variable, in this case, \(x\), alone in the center of the inequality. We started with the expression: \[-5 < -x - 8 < 5\]. We can isolate \(x\) by executing actions like adding, subtracting, multiplying, or dividing across every part of the inequality. Crucially, remember to reverse the inequality symbol when multiplying or dividing by a negative number. Here’s how we did it:

• Add 8 to each segment, moving towards isolating \(-x\).
• Multiply through by -1, flipping inequality signs, to solve for positive \(x\).

This process transforms the original inequality into a simpler form: \[-13 < x < -3\]. This technique applies broadly across algebraic problems ensuring the variable is cleanly isolated in its most readable position.