Problem 106 - A Graphical Approach to Precalculus with Limits

Question

Solve each compound inequality analytically. Support your answer graphically. $$4>6 x+5>-1$$

Step-by-Step Solution

Answer

The solution is \(-1 < x < \frac{-1}{6}\).

Step 1: Break Down the Compound Inequality

The compound inequality given is \( 4 > 6x + 5 > -1 \). This can be separated into two simple inequalities: \( 4 > 6x + 5 \) and \( 6x + 5 > -1 \).

Step 2: Solve the First Inequality

Solve \( 4 > 6x + 5 \): Subtract 5 from both sides: \[ 4 - 5 > 6x \]\[ -1 > 6x \]Divide both sides by 6: \[ \frac{-1}{6} > x \]This can be rewritten as \( x < \frac{-1}{6} \).

Step 3: Solve the Second Inequality

Solve \( 6x + 5 > -1 \): Subtract 5 from both sides: \[ 6x > -1 - 5 \]\[ 6x > -6 \]Divide both sides by 6: \[ x > -1 \]

Step 4: Combine the Solutions

The solutions to the two inequalities are \( x < \frac{-1}{6} \) and \( x > -1 \). Combining these gives the compound solution: \[ -1 < x < \frac{-1}{6} \].

Step 5: Graph the Solution on a Number Line

Graph the solution on a number line showing the interval \((-1, \frac{-1}{6})\). This means that \(x\) is between \(-1\) and \(\frac{-1}{6}\), not including \(-1\) and \( \frac{-1}{6} \). Draw an open circle on \(-1\) and \(\frac{-1}{6}\) to indicate these points are not included, and shade the region between them.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It helps algebra students express solutions to inequalities in a concise and clear manner. For compound inequalities like the one in our problem, interval notation is particularly useful.

In our solution, we found the inequality to be \(-1 < x < \frac{-1}{6}\). This describes the set of all numbers that are greater than \(-1\) and less than \(\frac{-1}{6}\). In interval notation, this set can be written as \((-1, \frac{-1}{6})\).

Here's how to interpret this notation:

• An open parenthesis \(()\) indicates the endpoint is not included in the interval. Thus, neither \(-1\) nor \(\frac{-1}{6}\) are part of the solution.
• When the interval does include an endpoint, square brackets \([]\) are used. For example, \([a, b]\) means "all numbers between \(a\) and \(b\), including \(a\) and \(b\)." In this particular problem, we don't have any square brackets since the endpoints are not part of the solution.

Knowing how to read and write interval notation transforms how you handle inequalities.

Graphical Solution

A graphical solution involves visually representing the solution of a compound inequality on a number line, which helps to improve understanding.

For the compound inequality \(-1 < x < \frac{-1}{6}\), you can draw a horizontal number line and mark the critical points of interest: \(-1\) and \(\frac{-1}{6}\). Since these two values are not included in the solution, we represent them with open circles.

Here are the steps to graph the solution:

• Draw a number line with clear markings for numbers between \(-2\) to \(0\) (to ensure you can easily see \(-1\) and \(\frac{-1}{6}\)).
• Place an open circle on \(-1\) and another on \(\frac{-1}{6}\).
• Shade the region between \(-1\) and \(\frac{-1}{6}\) to indicate all numbers in this range are solutions.

The open circles clearly show these boundary numbers are not included in the solution. This visual approach helps to reinforce the concept of the interval and provides an intuitive sense of the inequality's solution.

Analytical Solution

Analytical solutions involve using algebraic manipulations to find the solution to inequalities. For this exercise, the compound inequality \(4 > 6x + 5 > -1\) is split into two simpler inequalities which we solve separately.

First, solve \(4 > 6x + 5\):

• Subtract \(5\) from both sides to isolate the term with \(x\): \(4 - 5 > 6x\). Results in \(-1 > 6x\).
• Then, divide both sides by \(6\) to solve for \(x\): \(-\frac{1}{6} > x\), which can be rewritten as \(x < -\frac{1}{6}\).

Next, solve \(6x + 5 > -1\):

• Subtract \(5\) from both sides: \(6x > -1 - 5\), resulting in \(6x > -6\).
• Divide both sides by \(6\), yielding \(x > -1\).

Finally, combine the solutions for a comprehensive understanding. The solution \(x > -1\) and \(x < -\frac{1}{6}\) together form the interval \(-1 < x < -\frac{1}{6}\).
Understand these algebraic steps carefully, as they ensure precision and clarity when solving compound inequalities.