Problem 73 - Algebra and Trigonometry

Question

Solve: \(|(5 x-7)-5| \leq 0.05\)

Step-by-Step Solution

Answer

2.39 \leq x \leq 2.41

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|(5x - 7) - 5| \leq 0.05\). This inequality indicates that the absolute value of \((5x-7) - 5\) should be less than or equal to 0.05. To solve this, we must isolate \(5x\) and solve the resulting inequality.

Step 2: Simplify the Expression Inside the Absolute Value

Begin by simplifying the expression inside the absolute value: \((5x - 7) - 5\). This simplifies to \(5x - 12\). Thus, our inequality becomes \(|5x - 12| \leq 0.05\).

Step 3: Break Down the Absolute Value Inequality

Recall that \(|a| \leq b\) implies \(-b \leq a \leq b\). In our case, \(a = 5x - 12\) and \(b = 0.05\). So, we can write the inequality as: \(-0.05 \leq 5x - 12 \leq 0.05\).

Step 4: Solve the Compound Inequality

Solve the compound inequality \(-0.05 \leq 5x - 12 \leq 0.05\) by isolating \(x\):\[-0.05 + 12 \leq 5x \leq 0.05 + 12\]\[11.95 \leq 5x \leq 12.05\].

Step 5: Isolate the Variable

Divide all parts of the inequality \(11.95 \leq 5x \leq 12.05\) by 5 to isolate \(x\):\[ \frac{11.95}{5} \leq x \leq \frac{12.05}{5} \]\[2.39 \leq x \leq 2.41\].

Key Concepts

absolute value

Absolute value refers to the distance of a number from zero on the number line. It is always non-negative, because distance cannot be negative.
For any real number, the absolute value is denoted by vertical bars, like this: \(|a|\). For example, \(|3| = 3\) and \(|-3| = 3\).
When dealing with equations and inequalities, you often need to understand that \(|a| = b\) implies two cases: \(a = b\) or \(a = -b\).
In the context of solving inequalities, remembering this concept will help you break down complex expressions and simplify them.

inequalities

Inequalities are mathematical expressions that relate two values with symbols like <, >, ≤, ≥. They show that one value is larger or smaller than the other.
There are key properties and rules for inequalities:

• Adding or subtracting the same number on both sides keeps the inequality.
• Multiplying or dividing by a positive number keeps the inequality direction.
• Multiplying or dividing by a negative number reverses the inequality direction.

It's crucial to be careful with the inequality sign, especially when isolating variables.
This helps ensure the solution remains correct.

compound inequalities

Compound inequalities involve two separate inequalities joined by 'and' or 'or'.
When joined by 'and', both inequalities must be true simultaneously. For example, if \(-2 < x < 3\), then x must be greater than -2 and less than 3.
For 'or' compound inequalities, at least one of the inequalities must be true. For instance, if \(x < -2 \text{ or } x > 3\), x fits one of these conditions.
Breaking down an absolute value inequality often results in a compound inequality.
Understanding how to work with these helps solve more complex problems effectively.

solving linear equations

Solving linear equations involves finding the value of the variable that makes the equation true.
Here's a straightforward approach:

• Isolate the variable on one side of the equation.
• Use inverse operations to undo addition, subtraction, multiplication, or division.
• Simplify your results to get the final answer.

For example, in the exercise \(|5x - 12| \leq 0.05\), isolating x required dividing both sides of the inequality by 5.
Remember, the goal is to keep the equation balanced while isolating the variable.
Practice with these steps will make solving linear equations quicker and easier.