Problem 90 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(-11 x-(6 x-4)+5-3 x \leq 1\) (b) \(-11 x-(6 x-4)+5-3 x>1\)

Step-by-Step Solution

Answer

(a) [\frac{2}{5}, \infty), (b) (-\infty, \frac{2}{5})

Step 1: Simplify the Inequality for Part (a)

First, expand and simplify the inequality \(-11x-(6x-4)+5-3x \leq 1\).Start by distributing the minus sign: \(-11x - 6x + 4 + 5 - 3x \leq 1\).Combine like terms: \(-11x - 6x - 3x + 4 + 5 \leq 1\).Simplify further to get: \(-20x + 9 \leq 1\).

Step 2: Solve for x in Part (a)

To isolate \(x\), subtract 9 from both sides: \(-20x \leq 1 - 9\).This simplifies to: \(-20x \leq -8\).Now, divide both sides by \(-20\) and remember to flip the inequality sign:\(x \geq \frac{-8}{-20}\).This results in \(x \geq \frac{2}{5}\).

Step 3: Write the Solution in Interval Notation for Part (a)

The solution \(x \geq \frac{2}{5}\) means that \(x\) can be any value greater than or equal to \(\frac{2}{5}\).In interval notation, this is written as: \([\frac{2}{5}, \infty)\).

Step 4: Simplify and Solve the Inequality for Part (b)

For part (b), expand and simplify the inequality \(-11x-(6x-4)+5-3x > 1\).Following the same first step as (a): \(-20x + 9 > 1\).Subtract 9 from both sides: \(-20x > 1 - 9\).Which simplifies to: \(-20x > -8\).Divide both sides by \(-20\) and flip the inequality sign:\(x < \frac{2}{5}\).

Step 5: Write the Solution in Interval Notation for Part (b)

The solution \(x < \frac{2}{5}\) implies that \(x\) can be any value less than \(\frac{2}{5}\).In interval notation, this is represented as: \((-\infty, \frac{2}{5})\).

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers along a number line. It's often used with inequalities to show the range of possible solutions. In interval notation, parentheses \( ( ) \) denote that an endpoint is not included, whereas brackets \[ [ ] \] mean the endpoint is included.
For example:

• \((2, 5)\) means all numbers greater than 2 and less than 5, excluding 2 and 5 themselves.
• \([2, 5]\) includes every number from 2 to 5, including both.
• When we use \(-\infty\) or \(\infty\), these symbols always go with parentheses because infinity is a concept, not an actual number.

Interval notation is important when you solve inequalities because it provides a clear and concise way to express solution sets.
In our exercise, for Part (a), we found that the solution to the inequality is represented as \[ \left[ \frac{2}{5}, \infty \right) \], stating that all numbers from \( \frac{2}{5} \) onward satisfy the inequality. For Part (b), it is \( (-\infty, \frac{2}{5}) \), indicating all numbers less than \( \frac{2}{5} \) are solutions.

Solve Inequalities

Solving inequalities involves finding the value ranges for the variable that make the inequality true. It is similar to solving equations, but with important differences, especially when multiplying or dividing by negative numbers.
Here's a step-by-step approach:

• Simplify: Combine like terms and eliminate any unnecessary parts of the inequality.
• Isolate the Variable: Perform operations to get the variable on one side of the inequality.
• Direction of Inequality: Remember, multiplying or dividing by a negative number flips the inequality symbol. This can change when the solutions are for greater than (>) or less than (<).

In our exercise, starting with \(-20x + 9 \leq 1\) for Part (a), you subtract 9 from both sides and flip the inequality after dividing by \(-20\). Similarly, for Part (b), \(-20x + 9 > 1\), the steps are replicated, but the final inequality flips direction due to division by a negative number. This process ensures you correctly identify your solution's valid range.

Graphical Support

Graphical support can help understand inequalities better by visualizing solution sets on a number line or graph. The visual aspect makes it easier to see which portions of the number line satisfy the inequality.
Consider a number line graph:

• Open Circles: Use open circles on a number line for values that are not part of the solution set. These are typically for inequalities using \(<\) or \(>\).
• Closed Circles: Closed or filled circles indicate that the endpoint is part of the solution set. These correlate with \(\leq\) or \(\geq\).
• Shading: Shade the region of the number line representing all numbers in the solution set. This visually communicates which values satisfy the inequality.

For our inequalities, Part (a) would feature a closed circle at \(\frac{2}{5}\) and shading extending to the right, indicating all numbers greater than or equal to \(\frac{2}{5}\). Part (b) uses an open circle at \(\frac{2}{5}\) with shading to the left, representing all numbers less than \(\frac{2}{5}\). Visualization makes it clearer and faster to grasp the solution ranges.