Problem 49 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$\left|5 x+\frac{1}{2}\right|-2<5$$

Step-by-Step Solution

Answer

The combined solution for x is

Step 1: Isolate the Absolute Value Expression

Add 2 to both sides of the inequality to isolate the absolute value expression.

Step 2: Write the Combined Inequality

The combined inequality accounts for both the positive and negative scenarios of the absolute value expression. Thus, it can be written as:

Step 3: Solve the Combined Inequality

Combine the inequalities and solve for x in each.

Step 4: Combine the Solutions

Combine the solutions from the two inequalities.

Key Concepts

absolute value equations

Absolute value equations involve expressions within absolute value bars \(| \ |\). The absolute value of a number is its distance from zero on the number line, irrespective of direction. Thus, the absolute value of both -3 and 3 is 3. When dealing with equations or inequalities with absolute values, we have to consider two cases:

• The expression inside the absolute value is equal to the positive value of the other side.
• The expression inside the absolute value is equal to the negative value of the other side.

This concept helps us split the equation into two parts to solve more easily. Let's see how this applies in the exercise given: \(|5x+\frac{1}{2}|-2 < 5\). First, we isolate the absolute value by adding 2 to both sides:

\(|5x+\frac{1}{2}| < 7\).

Next, we consider both positive and negative scenarios to solve the inequality. This approach applies to all absolute value equations.

inequality solution steps

To solve absolute value inequalities, follow these four essential steps:

• Step 1: Isolate the Absolute Value Expression
  Adjust the inequality to make the absolute value expression alone on one side. In our example: \(|5x+\frac{1}{2}| < 7 \).
• Step 2: Write the Combined Inequality
  Interpret the absolute value inequality as a combined inequality. For \(|A| < B\), this translates to -B < A < B. Applying this to our problem: -7 < 5x + \frac{1}{2}< 7.
• Step 3: Solve the Combined Inequality
  Split the combined inequality and solve each part. For our scenario:
  \begin{cases} -(7) < 5x + \frac{1}{2} \ \ 5x + \frac{1}{2} < 7 \end{cases}\.
  We solve both:
  
  \-7 - \frac{1}{2} < 5x \rightarrow 5x > -7.5 \rightarrow x > -1.5\
  
  And:
  \5x + \frac{1}{2} < 7 \rightarrow 5x < 6.5 \rightarrow x < 1.3
• Step 4: Combine the Solutions
  Combine solutions from both parts and write down the final interval: \-1.5 < x < 1.3\.

The solution suggests the range of values for x.

precalculus fundamentals

Absolute value inequalities are vital in precalculus and build on the fundamentals learned in earlier math courses. They build understanding for:

• Inequalities and how solutions can represent ranges of values.
• Manipulating equations, including isolating and solving for variables.
• Understanding and applying properties of absolute values.

Grasping these concepts prepares students for more advanced math. Absolute value functions appear in different contexts, including real-world applications. For example, engineers use absolute values to express tolerances in measurements, and analysts use them in fields like economics to denote deviations. Solving these effectively hones analytical skills and deepens mathematical intuition.