Problem 123 - Algebra for College Students

Question

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=4 x-8,\) and the difference of \(y\) and 3 is less than 0.001.

Step-by-Step Solution

Answer

The interval is (2.74975, 2.75025).

Step 1: Set Up the Inequality

Given that the difference between y and 3 must be less than 0.001, write the inequality in terms of y: ewline |y - 3| < 0.001.

Step 2: Substitute the Expression for y

Substitute y with the given expression: ewline |4x - 8 - 3| < 0.001.

Step 3: Simplify the Inequality

Simplify the expression inside the absolute value:ewline |4x - 11| < 0.001.

Step 4: Remove the Absolute Value

To solve the absolute value inequality, break it into two separate inequalities: ewline -0.001 < 4x - 11 < 0.001.

Step 5: Solve for x

Solve the compound inequality for x by isolating x: ewline First inequality: -0.001 < 4x - 11 ewline Add 11 to both sides: ewline 10.999 < 4x ewline Divide by 4: ewline 2.74975 < x ewline Second inequality: 4x - 11 < 0.001 ewline Add 11 to both sides: ewline 4x < 11.001 ewline Divide by 4: ewline x < 2.75025.

Step 6: Write the Interval

Combine the results from step 5 to write the open interval for x: ewline The interval is (2.74975, 2.75025).

Key Concepts

inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. Instead of saying that two values are equal, they tell us that one value is either greater than, less than, greater than or equal to, or less than or equal to the other. In this exercise, we are specifically dealing with a type of inequality that uses the 'less than' symbol (<).
To solve these types of problems, we need to know how to manipulate the inequality to isolate the variable (in this case, x) on one side. This often requires inverse operations like addition, subtraction, multiplication, and division.
Rules to remember:

• Adding or subtracting the same value on both sides of the inequality does not change the inequality's direction.
• Multiplying or dividing both sides by a positive number keeps the direction the same.
• Multiplying or dividing both sides by a negative number reverses the inequality's direction.

Understanding these rules helps simplify and solve the inequalities effectively.

absolute value

The absolute value of a number is its distance from zero on a number line, regardless of direction. It is always a non-negative value. When dealing with equations involving absolute values, like \(|y-3| < 0.001\), we learn that the absolute value creates a range within which the expression can vary.
The general approach involves:

• Writing the expression inside the absolute value as both negative and positive.
• Creating two separate inequalities from the expression.
• Solving each inequality independently.

In the given exercise, we start with |4x - 11| < 0.001. We then break it down into two separate inequalities:

• -0.001 < 4x - 11
• 4x - 11 < 0.001

This step is crucial. It allows us to handle a compound inequality in our calculations.

compound inequalities

Compound inequalities involve two separate inequalities that are combined into one statement by the words 'and' or 'or'. The goal is to find the values of the variable that satisfy both parts of the compound inequality if 'and' is used, or at least one part if 'or' is used.
In this exercise, after breaking the absolute value into two inequalities:

• -0.001 < 4x - 11
• 4x - 11 < 0.001

We solve each inequality separately and then combine the results. Solving compound inequalities involves:

• Isolating the variable in both inequalities.
• Combining the solutions to find the interval.

Here, after solving both inequalities, we combine the results to get the interval (2.74975, 2.75025), which tells us where x lies so the original condition is satisfied.

algebra techniques

Solving inequalities and absolute value problems often involves important algebra techniques. These include simplifying expressions, isolating variables, and handling compound inequalities.
Here's a breakdown of the steps used in our exercise:

• Setting up the inequality: Begin by expressing the problem in a form that can be solved.
• Substituting expressions: Replace variables with the given values or other expressions.
• Simplifying: Combine like terms and simplify the inequality.
• Splitting absolute values: Break them into two simpler inequalities.
• Solving: Use addition, subtraction, multiplication, or division to isolate the variable.
• Combining results: From both inequalities to form a final, single solution.

These algebraic methods aim to make the problem more manageable. Practicing these techniques helps build a strong foundation for more complex math problems.