Problem 58 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$|4-3 x|>1$$

Step-by-Step Solution

Answer

The set of solutions is \(x < 1\) or \(x > \frac{5}{3}\).

Step 1: Understand Absolute Value Inequality

The expression \(|a| > b\) means that the expression inside the absolute value \(a\) is either greater than \(b\) or less than \(-b\). This can be written as two separate inequalities: \(a > b\) or \(a < -b\).

Step 2: Break Down the Inequality

For the inequality \(|4 - 3x| > 1\), we get two separate inequalities: 1. \(4 - 3x > 1\)2. \(4 - 3x < -1\).

Step 3: Solve the First Inequality

Solve the inequality \(4 - 3x > 1\):Subtract 4 from both sides: \(-3x > 1 - 4\) or \(-3x > -3\)Divide both sides by \(-3\), remembering to reverse the inequality sign:\(x < 1\).

Step 4: Solve the Second Inequality

Solve the inequality \(4 - 3x < -1\):Subtract 4 from both sides:\(-3x < -1 - 4\) or \(-3x < -5\)Divide both sides by \(-3\), remembering to reverse the inequality sign:\(x > \frac{5}{3}\).

Step 5: Combine the Solutions

The solutions to the inequalities are combined to form \(x < 1\) or \(x > \frac{5}{3}\). This means that any value for \(x\) outside the interval \([\frac{5}{3}, 1]\) is a solution to the inequality.

Key Concepts

Solving Inequalities

When solving absolute value inequalities, it's important to understand what the absolute value actually represents. The absolute value of a number is its non-negative value, regardless of its original sign. Therefore, solving an inequality like \[|a| > b\] involves considering two possible cases:

• The expression inside the absolute value is greater than the number on the right.
• Or, the expression inside is less than the negative of that number.

For example, with the inequality \[|4 - 3x| > 1\], we're looking at two separate inequalities:

• \(4 - 3x > 1\)
• \(4 - 3x < -1\)

Once these inequalities are established, you tackle each one separately before combining their solutions for the final answer. This step-by-step approach ensures you address all possibilities for the variable involved. Remember, when an inequality includes an absolute value, you're essentially solving for conditions on both extremes.

Algebraic Expressions

Algebraic expressions form the foundation of problems involving inequalities. An algebraic expression comprises constants, variables, and operations like addition, subtraction, multiplication, and division. In our problem, \[ |4 - 3x| > 1 \],the expression inside the absolute value, \[4 - 3x\],is made of constant number 4 and variable term \(-3x\). Dealing with these expressions requires a keen understanding of algebraic principles.
To manipulate such expressions effectively, we often perform operations like:

• Adding or subtracting the same value on both sides of the equation to isolate terms.
• Multiplying or dividing by a negative number, which flips the inequality sign.

Care must be taken, especially when multiplying or dividing by negative numbers, to ensure the inequality remains accurate. This intricacy can lead to common pitfalls, but keeping rules in mind simplifies the process. With practice, handling algebraic expressions becomes intuitive, empowering you to solve complex inequalities with confidence.

Combining Solutions

After solving the individual inequalities, the next step is to combine the solutions. This process ensures we account for all possible values of the variable that satisfy the original inequality. For the inequality \[|4 - 3x| > 1\], the solutions to the separate inequalities were:

• \(x < 1\)
• \(x > \frac{5}{3}\)

When combining these results, we find that the solution set includes values less than 1 and greater than \(\frac{5}{3}\), but not those in between. This indicates the solution is the union of two intervals:

• The interval from \(-\infty\) to 1.
• The interval from \(\frac{5}{3}\) to \(+\infty\).

Graphically, this appears as two separate rays on a number line, illustrating where the solutions lie. The essence of combining solutions lies in visualizing or plotting the values to comprehensively capture the entire solution space. This ensures the interpretation of results aligns with the real-world context emphasized in algebra.