Problem 56 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$|7-4 x| \leq 11$$

Step-by-Step Solution

Answer

The solution is \(-1 \leq x \leq 4.5\).

Step 1: Understanding Absolute Inequalities

The inequality \(|7-4x| \leq 11\) involves an absolute value, which indicates the distance from zero on a number line. We need to consider separate cases where the expression inside the absolute value is both positive and negative because absolute values account for both.

Step 2: Setting up the Two Cases

The statement \(|A| \leq B\) translates to two inequalities: 1. \(A \leq B\)2. \(-A \leq B\)For our problem, that means:1. \(7 - 4x \leq 11\)2. \(-(7 - 4x) \leq 11\)

Step 3: Solving the First Inequality

Solve the first case equation:\[7 - 4x \leq 11\]Subtract 7 from both sides:\[-4x \leq 4\]Divide by -4 (remember to flip the inequality sign):\[x \geq -1\]

Step 4: Solving the Second Inequality

Now solve the second case equation:\[-(7 - 4x) \leq 11\]First, distribute the negative sign:\[-7 + 4x \leq 11\]Add 7 to both sides:\[4x \leq 18\]Divide both sides by 4:\[x \leq 4.5\]

Step 5: Combining the Solutions

Now we combine the solutions of \(x \geq -1\) and \(x \leq 4.5\). The solution to the inequality \(|7-4x| \leq 11\) is the intersection of these two solution sets:\[-1 \leq x \leq 4.5\]

Key Concepts

absolute value

When dealing with absolute values, we're essentially trying to measure the distance of a number from zero on the number line. The absolute value of a number is always non-negative. This means it doesn't matter if the number inside is negative or positive; the result will always be positive. For example, both \(|-3|\) and \(|3|\) give us 3.
In an inequality such as \(|7 - 4x| \leq 11\), the expression inside the absolute value (in this case, \(7 - 4x\)) could be either negative or positive. Therefore, we must evaluate both scenarios to find the full range of solutions.
Understanding this concept is crucial because it helps us break down the problem into simpler parts. We consider two separate inequalities: one where the expression inside is non-negative (\(7 - 4x \leq 11\)), and one where it's negative (\(-(7 - 4x) \leq 11\)). This dual-case approach helps us handle the complexity of absolute value inequalities.

inequality solving

Solving inequalities involves finding the range of values for the variable that makes the inequality true. Unlike equations, inequalities can have infinite solutions because they show a range of values rather than a specific one.
To solve an inequality, such as \(7 - 4x \leq 11\), we follow similar steps to solving equations: isolating the variable on one side. However, one crucial difference is the rule about flipping inequality signs. When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
Let's break down how we solve the inequalities from our example:

• First, solve \(7 - 4x \leq 11\) by subtracting 7 and then dividing by -4. Remember to flip the sign, leading to \(x \geq -1\).
• Next, solve \( -(7 - 4x) \leq 11\) by first distributing the negative, then solving in a similar manner, yielding \(x \leq 4.5\).

By knowing how to solve each part independently, we combine them to find the common solutions: \(-1 \leq x \leq 4.5\). This final range is the set of all numbers that satisfy the original absolute inequality.

number line interpretation

The number line is a fantastic tool for visualizing solutions to inequalities and equations, especially when dealing with absolute values. By plotting these inequalities, we get a clear picture of the range of values for which the inequality holds. For the solution \(-1 \leq x \leq 4.5\), a number line helps us visualize the interval between \(-1\) and \(4.5\). We start by marking two points, \(-1\) and \(4.5\), on the line. Then, shading the space between these points shows the solution set visually.
The number line interpretation helps by:

• Demonstrating where the solution begins and ends.
• Providing a concise picture of whether endpoints are included. In our case, both \(-1\) and \(4.5\) are included, denoted by closed circles.
• Helping students understand the concept of intervals in mathematics by offering a clear, visual method to evaluate and understand the inequalities.

This visual aid enhances our understanding of the solution, making it easier to grasp abstract concepts in inequality solving.