Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Solve each group of equations and inequalities analytically. (a) \(|2 x+1|+3=5\) (b) \(|2 x+1|+3 \leq 5\) (c) \(|2 x+1|+3 \geq 5\)

Step-by-Step Solution

Answer

(a) \(x = \frac{1}{2}\) or \(x = -\frac{3}{2}\); (b) \(-\frac{3}{2} \leq x \leq \frac{1}{2}\); (c) \(x \leq -\frac{3}{2}\) or \(x \geq \frac{1}{2}\).

Step 1: Isolate the Absolute Value (Equation)

First, subtract 3 from both sides of the equation \(|2x + 1| + 3 = 5\) to isolate the absolute value expression: \(|2x + 1| = 2\).

Step 2: Solve the Absolute Value Equation

The equation \(|2x + 1| = 2\) splits into two possible equations: 1. \(2x + 1 = 2\)2. \(2x + 1 = -2\). Solve both equations.For \(2x + 1 = 2\):- Subtract 1: \(2x = 1\) - Divide by 2: \(x = \frac{1}{2}\)For \(2x + 1 = -2\):- Subtract 1: \(2x = -3\) - Divide by 2: \(x = -\frac{3}{2}\).

Step 3: Isolate the Absolute Value (Inequality 1)

Subtract 3 from both sides of the inequality \(|2x + 1| + 3 \leq 5\) to get \(|2x + 1| \leq 2\).

Step 4: Solve the Absolute Value Inequality (1)

The inequality \(|2x + 1| \leq 2\) translates into a compound inequality: \(-2 \leq 2x + 1 \leq 2\). Solve it step-by-step:1. Subtract 1 from all parts: \(-3 \leq 2x \leq 1\)2. Divide by 2: \(-\frac{3}{2} \leq x \leq \frac{1}{2}\).

Step 5: Isolate the Absolute Value (Inequality 2)

Subtract 3 from both sides of the inequality \(|2x + 1| + 3 \geq 5\) to get \(|2x + 1| \geq 2\).

Step 6: Solve the Absolute Value Inequality (2)

The inequality \(|2x + 1| \geq 2\) splits into two separate inequalities:1. \(2x + 1 \geq 2\) 2. \(2x + 1 \leq -2\).Solve both separately:For \(2x + 1 \geq 2\):- Subtract 1: \(2x \geq 1\)- Divide by 2: \(x \geq \frac{1}{2}\)For \(2x + 1 \leq -2\):- Subtract 1: \(2x \leq -3\)- Divide by 2: \(x \leq -\frac{3}{2}\).

Step 7: Summary of Solutions

For part (a), the solutions are \(x = \frac{1}{2}\) or \(x = -\frac{3}{2}\). For(b), the solution interval is \(-\frac{3}{2} \leq x \leq \frac{1}{2}\). For (c),the solution is the union of two intervals: \(x \leq -\frac{3}{2}\) or \(x \geq \frac{1}{2}\).

Key Concepts

Understanding Inequalities in Absolute Values

Inequalities in mathematics are expressions that indicate one quantity is larger or smaller than another. When it comes to absolute value inequalities, they involve comparing the difference between numbers, taking only their magnitude without regard to their direction on the number line.
In the context of our exercise, we transformed absolute value expressions into inequalities. For example:

• The inequality \(|2x + 1| + 3 \leq 5\) becomes \(|2x + 1| \leq 2\) after isolating the absolute value.
• Then, it can be solved as a compound inequality: - This means considering two cases simultaneously. - That’s because the values of \(2x + 1\) must fall within -2 and 2.

These cases of inequalities help us solve problems by setting boundaries and understanding which values of \(x\) satisfy the inequality condition.

Solving Compound Inequalities

Compound inequalities can be thought of as two inequalities combined into one. These inequalities broaden the range of values for which the inequality holds true.
In the exercise, after simplifying the inequality \(|2x + 1| \leq 2\), we had:

• \(-2 \leq 2x + 1 \leq 2\)

This is a compound inequality. It represents a range where the expression \(2x + 1\) is greater than or equal to -2 and at the same time less than or equal to 2. As a solution:

• We first subtract 1 from each part: \(-3 \leq 2x \leq 1\).
• After isolating \(x\) by dividing by 2, we got: \(-\frac{3}{2} \leq x \leq \frac{1}{2}\).

Here, \(x\) is neatly confined within a specific range, which satisfies the inequality. Such ranges are very useful in graphical and real-world scenarios where variables must be kept in check.

Isolation of Variables in Absolute Value Situations

The process of isolating a variable is crucial in solving equations and inequalities, especially those involving absolute values. It involves rearranging the equation so that the variable of interest stands alone on one side of the equality.
In our exercises, we isolated the absolute value by:

• Subtracting constants from both sides to simplify the equation or inequality.

For example, in \(|2x + 1| + 3 = 5\), subtracting 3 from both sides provides \(|2x + 1| = 2\). This isolation step prepares us to either solve two separate equations or form compound inequalities. By executing these steps, we can effectively deal with more complex scenarios:

• Absolute value equations become straightforward linear equations.
• Absolute value inequalities form composite scenarios which we compromise for balanced outcomes.

Mastering isolation allows you to simplify and solve even the most daunting-looking equations and inequalities swiftly.