Problem 122 - Algebra for College Students

Question

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=4 x-6,\) and the difference of \(y\) and 2 is less than 0.02.

Step-by-Step Solution

Answer

1.995 < x < 2.005

Step 1: Identify the given condition

The equation given is: y = 4x - 6 The problem states that the difference between y and 2 is less than 0.02. Mathematically, this is expressed as: |y - 2| < 0.02

Step 2: Substitute y into the inequality

We know that y = 4x - 6. Substitute y in the equation |y - 2| < 0.02: |(4x - 6) - 2| < 0.02 Simplify the expression inside the absolute value: |4x - 8| < 0.02

Step 3: Solve the absolute value inequality

To solve |4x - 8| < 0.02, we split it into two cases: 1. 4x - 8 < 0.02 2. 4x - 8 > -0.02 Solve each inequality separately: 4x - 8 < 0.02 4x < 8.02 x < 2.005 And: 4x - 8 > -0.02 4x > 7.98 x > 1.995

Step 4: Combine the intervals

Combine the two inequalities: 1.995 < x < 2.005 This is the open interval in which x must lie.

Key Concepts

solving inequalities

Inequalities are like balancing acts. Instead of saying things are equal, they show a range of possibilities. If you have an inequality like:
4x - 8 < 0.02, it means the expression 4x - 8 must be less than 0.02.
To solve such inequalities, perform operations like adding, subtracting, multiplying, or dividing, just like you do in normal equations. However, pay attention when multiplying or dividing by negative numbers - the inequality sign flips! For example, dividing both sides by 4 in 4x < 8.02 gives:
x < 2.005.

absolute value

The absolute value of a number is a way to describe its distance from zero on a number line.
It's always positive or zero. When you see:
|a| < b, understand it as:
-a < a < b. In the exercise, we had:
|4x - 8| < 0.02. This means:
-0.02 < 4x - 8 < 0.02.

inequality intervals

After solving separate inequalities, you often combine them into a single interval. This gives a range showing all possible values.
From the exercise, we got two refined inequalities:
x < 2.005 and x > 1.995. These can be combined into the interval:
1.995 < x < 2.005, signifying that x can be any number between 1.995 and 2.005. Notice these small open intervals might be harder to identify, but practice makes perfect!

linear equations

Linear equations are straightforward. They look like: y = mx + b, where m and b are constants. In our exercise, y = 4x - 6. These types of equations graph as straight lines. By substituting y = 4x - 6 into the inequality |y - 2| < 0.02, we could analyze the problem. This approach helps with visual understanding too! Be comfortable with manipulating such equations, adding, subtracting constants, or even changing forms. Linear methods build foundations for more advanced math topics later.