Problem 87 - 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) \(10 x+5-7 x \geq 8(x+2)+4\) (b) \(10 x+5-7 x<8(x+2)+4\)

Step-by-Step Solution

Answer

(a) \((-\infty, -3]\); (b) \((-3, \infty)\).

Step 1: Simplify Both Sides (Inequality a)

Start by simplifying both sides of the inequality. The given inequality is \(10x + 5 - 7x \geq 8(x + 2) + 4\). First, combine like terms on the left: \(3x + 5 \geq 8(x + 2) + 4\). Then distribute the 8 on the right: \(3x + 5 \geq 8x + 16 + 4\). This simplifies further to \(3x + 5 \geq 8x + 20\).

Step 2: Isolate the Variable (Inequality a)

Move all terms involving \(x\) to one side and constant terms to the other. Subtract \(3x\) from both sides: \(5 \geq 5x + 20\). Then subtract 20: \(5 - 20 \geq 5x\). Simplify this to \(-15 \geq 5x\).

Step 3: Solve for x (Inequality a)

To isolate \(x\), divide both sides by 5: \(-3 \geq x\) or equivalently \(x \leq -3\). The solution set in interval notation is \((-\infty, -3]\).

Step 4: Simplify and Solve (Inequality b)

For inequality b, \(10x + 5 - 7x < 8(x + 2) + 4\), follow the same step as in the first inequality. Simplify to \(3x + 5 < 8x + 20\). Isolate \(x\) by subtracting \(3x\) and then 20, giving \(-15 < 5x\). Divide by 5: \(-3 < x\) or equivalently \(x > -3\). The solution set in interval notation is \((-3, \infty)\).

Step 5: Graph the Solution

Represent the solutions graphically on a number line. For \(x \leq -3\), draw a solid line to the left with a closed dot at -3. For \(x > -3\), draw a line to the right with an open dot at -3.

Key Concepts

Graphical Representation of Inequalities

Graphical representation is a powerful way to visualize inequalities and understand their solutions. To graphically represent an inequality, we consider a number line where each point represents a potential value for the variable. For the inequality solution, decide whether to use open or closed dots:

• Closed dot: Used when the number is included in the solution (e.g., when using "\( \leq \)" or "\( \geq \)" in the inequality).
• Open dot: Used when the number is not included in the solution (e.g., for strict inequalities "\( < \)" or "\( > \)").

In our exercise, the solution \( x \leq -3 \) is represented by a closed dot on -3 and shading to the left, indicating that \( x \) can be any number less than or equal to -3. Conversely, for \( x > -3 \), an open dot at -3 and shading to the right shows that \( x \) can be any number greater than -3.
This visual representation helps solidify our grasp of which values the variable can take and correlates well with the solution in interval notation.

Interval Notation for Solutions

Interval notation offers a systematic way to express the solution set of inequalities, providing clarity and consistency. Our solutions often involve either closed intervals, open intervals, or a combination of both:

• Closed interval: Noted with square brackets \([a, b]\), includes both \(a\) and \(b\).
• Open interval: Noted with parentheses \((a, b)\), which excludes \(a\) and \(b\).
• Half-open interval: Combines a parenthesis and a bracket, such as \((a, b]\) or \([a, b)\).

In the given exercise, the solution \( x \leq -3 \) is expressed in interval notation as \((-\infty, -3]\), showing that the solution includes all values less than or equal to -3, extending indefinitely towards negative infinity. Similarly, \( x > -3 \) becomes \((-3, \infty)\), which includes all values greater than -3, indefinitely to positive infinity. This concise notation aids in quickly identifying solution sets and communicating them effectively.

Algebraic Manipulation of Inequalities

Algebraic manipulation is a crucial skill when solving inequalities. It involves simplifying expressions and isolating the variable to determine the solution set. Let's break it down:
First, start by simplifying each side of the inequality. Identify like terms and combine them. For example, from the inequality part (a), simplify \(10x + 5 - 7x \) to \(3x + 5\). Then, distribute and simplify any expressions on the opposite side.
Next, focus on isolating the variable. Shift all terms containing the variable to one side and constants to the other. For instance, moving \(3x\) away from the left side results in \(5 \geq 5x + 20\). Subsequently, solve by performing arithmetic operations such as addition, subtraction, or division.
Finally, when the variable is isolated, rewrite the inequality to reflect the solution. In our example for part (a), divide \(-15 \geq 5x\) by 5 to get \(x \leq -3\). This process not only solves the inequality but also ensures that each transformation step is reversible, maintaining the balance of the equation throughout.