Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation graphically. $$|2 x+2|+|x+1|=9$$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understanding Absolute Value Equations

The equation \(|2x+2| + |x+1| = 9\) involves absolute values, which means we need to consider different cases based on when each expression inside the absolute value becomes zero.

Step 2: Identify Critical Points

The critical points occur when the expressions inside the absolute values are zero: \(2x+2=0\) and \(x+1=0\). Solving these gives \(x = -1\) and \(x = -0.5\). These points divide the number line into three intervals to consider: \((-\infty, -1)\), \((-1, -0.5)\), and \((-0.5, \infty)\).

Step 3: Solve for Each Interval

For each interval, solve the equation considering the sign of each expression:- **Interval \((-\infty, -1)\)**: Both expressions are negative, rewrite equation as \(-2x-2 - x - 1 = 9\). Simplify to find no solution as it leads to a contradiction. - **Interval \((-1, -0.5)\)**: The first is positive, the second is negative, rewrite equation as \(2x+2 - (x+1) = 9\). Simplify to find no solution as it leads to a smaller sum.- **Interval \((-0.5, \infty)\)**: Both are positive, rewrite equation as \(2x+2 + x + 1 = 9\). Simplify to \(3x + 3 = 9\), leading to \(x = 2\).

Step 4: Graphical Interpretation

Plot the functions \(y_1 = |2x+2| + |x+1|\) and \(y_2 = 9\) on the same axes. Identify the intersection points of these two graphs. Based on our analysis, \(y_1\) is at most \(9\) only on the interval \((-0.5, \infty)\). Verify by the graph that the intersection occurs at \(x = 2\).

Step 5: Conclusion

The graphical solution confirms the equation is satisfied when \(x = 2\). Therefore, the solution to the equation \(|2x+2| + |x+1| = 9\) is \(x = 2\).

Key Concepts

Absolute Value

The concept of absolute value is essential in understanding this type of equation. Absolute value refers to the distance of a number from zero on the number line, without considering direction. For instance, \(|a|\) represents the absolute value of \(a\), which is equal to \(a\) if \(a\) is positive or zero, and \(-a\) if \(a\) is negative. This property means that each absolute value variable can change its behavior depending on the sign of the expression inside it.

When dealing with absolute value equations, your goal is to identify when and how each expression changes as you move across different sections of the number line. By focusing on the points where expressions inside absolute values equal zero, you'll be able to break down the equation into manageable cases, each defined by a different interval.

Graphical Solution

Graphical solutions provide a visual representation of an equation, helping to pinpoint where exact solutions occur. For the equation \(|2x+2| + |x+1| = 9\), plotting the function \(y_1 = |2x+2| + |x+1|\) can give insights into its behavior over all possible values of \(x\).

To solve graphically:

• Draw the functions \(y_1\) and \(y_2 = 9\) on the same graph.
• Identify where these graphs intersect - these points provide possible solutions.
• The intersection occurs at \(x = 2\), indicating the solution to the equation.

Graphical solutions are very effective for visualizing solutions but require understanding the key aspects of graph behavior, such as increasing, decreasing, and the effect of absolute values on graph shape.

Critical Points

Critical points are vital in solving absolute value equations, as they split the number line into distinct intervals where the behavior of the absolute value changes. These points are determined by setting each expression inside the absolute value equal to zero.

In our example:

• For \(|2x+2|\), solve \(2x+2=0\) to get \(x = -1\).
• For \(|x+1|\), solve \(x+1=0\) to get \(x = -0.5\).

These critical points divide the number line into three intervals:

• \((-\infty, -1)\) - both expressions negative.
• \((-1, -0.5)\) - first positive, second negative.
• \((-0.5, \infty)\) - both expressions positive.

By analyzing these intervals separately, one can determine how the equation should be adjusted for each scenario. This step is crucial for an accurate solution.

Equation Solving

To solve the equation \(|2x+2| + |x+1| = 9\), you need to handle each interval created by the critical points with care. This involves rewriting the equation based on the signs of the expressions within these intervals:

• **Interval\((-\infty, -1)\)**: Transform the equation to \(-2x-2 - x - 1 = 9\). This leads to a contradiction, eliminating solutions in this range.
• **Interval \((-1, -0.5)\)**: Transform to \(2x+2 - (x+1) = 9\). Again, this results in a false equation, ruling out solutions here.
• **Interval \((-0.5, \infty)\)**: Substitute with \(2x+2 + x + 1 = 9\), simplifying to \(3x + 3 = 9\). Solving gives \(x = 2\), an actual solution.

Equation solving involves detailed examination of each interval, ensuring every possible scenario is tested to confirm valid solutions. Checking your results with a graphical method further validates these findings.