Problem 68 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ \left|\frac{3(x-1)}{4}\right|<6 $$

Step-by-Step Solution

Answer

The solution to the inequality \( \left| \frac{3(x-1)}{4} \right| < 6 \) is \( -7 < x < 9 \).

Step 1: Understand the propoerties of absolute value

Recall that for any real number a, \( |a| < b \) if and only if \( -b < a < b \). We can apply this property to the given inequality.

Step 2: Convert the absolute value inequality into a compound inequality

Therefore, \( \left| \frac{3(x-1)}{4} \right| < 6 \) is equivalent to \( -6 < \frac{3(x-1)}{4} < 6 \).

Step 3: Solve the compound inequality

First multiply each part of the inequality by 4 to clear the fraction, which gives \( -24 < 3(x-1) < 24 \). Then, divide each part of the inequality by 3 to solve for x, which gives the solution: \( -8 < x-1 < 8 \). Last, add 1 to each part of the inequality to solve for x, giving the final solution as \( -7 < x < 9 \).

Key Concepts

Understanding Compound Inequalities

Compound inequalities combine two inequalities into one statement, connecting them with either 'and' or 'or'. In the context of absolute value, these are used to express a range of values. For example, if you have an absolute value inequality like \(|x| < 6\), it means the distance of \(x\) from zero is less than 6. This translates to a compound inequality:

• \(-6 < x < 6\)

This shows that \(x\) can be any number between -6 and 6. For the problem in the exercise, converting the absolute value into a compound inequality lets you work within defined boundaries.
Solving these involves treating each part separately, ensuring the solutions satisfy both conditions. Compound inequalities are crucial for dealing with ranges of values in real-world scenarios, making them a vital algebraic concept.

Solving Inequalities Step-by-Step

Solving inequalities involves finding what values make an inequality true. You approach it by isolating the variable, just like with equations, but pay attention to the direction of the inequality sign. Special care is needed when multiplying or dividing by negative numbers, as this reverses the inequality sign.
In our exercise, once we set up the compound inequality \(-6 < \frac{3(x-1)}{4} < 6\), we first clear the fraction by multiplying every part by 4:

• \(-24 < 3(x-1) < 24\)

Next, divide every part by 3 to further simplify:

• \(-8 < x-1 < 8\)

Finally, isolate \(x\) by adding 1 to each part:

• \(-7 < x < 9\)

This tells us that x can be any number between -7 and 9.
By understanding each step, you can systematically solve and verify your inequalities.

Exploring Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In inequalities, understanding these expressions is essential to manipulating and simplifying them.
The exercise features an expression \(\frac{3(x-1)}{4}\), which includes the process of arithmetic operations linked through variables. Here’s how to understand the pieces:

• The term \(3(x-1)\) shows multiplication and distribution applied to \(x\).
• Dividing this product by 4 creates a fractional expression, which needs manipulation to simplify.

Clear these fractions early in solving by multiplying through the entire inequality. Remember, consistent application across all parts of the inequality keeps solutions valid.
Learning how to navigate and manipulate algebraic expressions is a powerful tool in algebra, of which solving absolute value inequalities is just one application.