Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality graphically. $$-|3 x-12| \geq-x-1$$

Step-by-Step Solution

Answer

The solution is \([2.75, 6.5]\).

Step 1: Simplify the Inequality

Begin by simplifying the inequality \(-|3x - 12| \geq -x - 1\). First, get rid of the negative signs by multiplying the entire inequality by -1, which reverses the inequality sign: \(|3x - 12| \leq x + 1\). This new inequality suggests that we are looking for the intersection of two conditions due to the absolute value.

Step 2: Break Down the Absolute Value

The inequality \(|3x - 12| \leq x + 1\) implies two conditions need to be solved separately: 1. \(3x - 12 \leq x + 1\)2. \(3x - 12 \geq -(x + 1)\). These conditions stem from the absolute value expression, which can be positive or negative.

Step 3: Solve the First Condition

Solve the inequality \(3x - 12 \leq x + 1\). To isolate \(x\), subtract \(x\) from both sides: \(2x - 12 \leq 1\). Add 12 to both sides: \(2x \leq 13\). Finally, divide by 2: \(x \leq 6.5\).

Step 4: Solve the Second Condition

Solve the inequality \(3x - 12 \geq -(x + 1)\). First, distribute the negative sign: \(3x - 12 \geq -x - 1\). Add \(x\) to both sides: \(4x - 12 \geq -1\). Add 12 to both sides: \(4x \geq 11\). Divide by 4: \(x \geq 2.75\).

Step 5: Graph the Solution

Graph both conditions on the number line: 1. \(x \leq 6.5\) selects all values left of and including 6.5.2. \(x \geq 2.75\) selects values right of and including 2.75.The solution is the interval where both conditions overlap: \([2.75, 6.5]\).

Key Concepts

Solving Inequalities

Inequalities let us express a wide range of conditions mathematically. In this problem, we are given an inequality involving an absolute value, which typically results in multiple conditions to explore. When solving inequalities, one fundamental rule is that multiplying or dividing by a negative number requires us to reverse the inequality sign. This ensures that the inequality maintains its correct relationship.

To tackle these problems efficiently, follow these general steps:

• Simplify the inequality as much as possible by using basic algebraic operations.
• Handle operations involving negative signs carefully, especially when multiplying or dividing.
• Break down complex inequalities into simpler components, especially if absolute values are involved, generating multiple cases or conditions that need solving.
• Graphing or sketching may help visualize the solution range.

Understanding these foundational concepts can greatly enhance your ability to solve more complex inequality-related problems.

Absolute Value Equations

Absolute value equations require special attention because they consider both positive and negative possibilities. The absolute value of a number is the distance from zero, unaffected by direction. When dealing with an equation of the form \(|ax + b| \leq c\), where a, b, and c are constants, we are actually looking at two separate inequalities:

• \( ax + b \leq c \)
• \( ax + b \geq -c \)

After solving these two inequalities separately, you combine the solutions to find where they overlap, providing the full set of values that satisfy the original absolute value inequality.

In the initial exercise, we recognized that the inequality was simplified by removing negative signs and shifting terms, resulting in the absolute value form. Solving it led to two separate inequalities, which when graphed, revealed the intersection as the solution. Understanding the dual nature of absolute value expressions is crucial in obtaining the correct solutions.

Number Line Graphing

Graphing on a number line is a visual method to represent solutions of equations and inequalities. It helps us understand which parts of the number line satisfy the given conditions. In our solved problem, number line graphing demonstrated how to visualize the overlapping range of solutions.

Here's how to graph these inequalities effectively:

• Mark the points where each inequality is equal, often these are the endpoints (e.g., \(x = 6.5\) and \(x = 2.75\)).
• Shading or curving above the number line between these points shows the range where both conditions are true simultaneously.
• For inequalities involving "greater than or equal to" or "less than or equal to," use closed circles on endpoints include them in the solution. Open circles signify non-inclusion if it's just "greater" or "less."

This intuitive visual can make complex algebra feel more approachable. Number line graphing becomes particularly helpful when combining solutions from multiple inequalities, as it immediately gives a clear picture of where solutions overlap and which intervals satisfy the problem.