Problem 58 - Elementary and Intermediate Algebra

Question

Solve each compound inequality. Graph the solution set and write it using interval notation. $$ \frac{7}{3} x+2 \leq 16 \text { and }-8 x \geq-48 $$

Step-by-Step Solution

Answer

The solution is \( x \leq 6 \) or interval notation \(( -\infty, 6 ]\).

Step 1: Solve the First Inequality

The first inequality is \( \frac{7}{3}x + 2 \leq 16 \). To solve for \( x \), first subtract 2 from both sides to get \( \frac{7}{3}x \leq 14 \). Then multiply both sides by \( \frac{3}{7} \) to isolate \( x \), resulting in \( x \leq 6 \).

Step 2: Solve the Second Inequality

The second inequality is \( -8x \geq -48 \). Start by dividing both sides by \( -8 \). Remember to reverse the inequality sign when dividing by a negative number. This gives \( x \leq 6 \).

Step 3: Combine the Solutions

Both inequalities lead to the same expression \( x \leq 6 \). Thus, the compound inequality solution is \( x \leq 6 \).

Step 4: Represent the Solution on a Number Line

The solution \( x \leq 6 \) is represented on a number line by shading to the left of 6, including 6 itself, which is indicated by a solid dot at 6.

Step 5: Write the Solution in Interval Notation

The solution set \( x \leq 6 \) can be written in interval notation as \(( -\infty, 6 ]\).

Key Concepts

Inequality Solving

Inequality solving involves finding the range of values that satisfy a given inequality. In the compound inequality exercise, we have two separate inequalities: \( \frac{7}{3}x + 2 \leq 16 \) and \( -8x \geq -48 \). Our goal is to find the values of \( x \) that satisfy both simultaneously.
**Steps to Solve an Inequality:**

• Isolate the Variable: Aim to get \( x \) by itself on one side of the inequality.
• Perform Legal Operations: You can add, subtract, multiply, or divide both sides by the same number. However, remember, if you multiply or divide by a negative number, the direction of the inequality symbol must be reversed.
• Combine Solutions: For compound inequalities, ensure the solution meets all conditions given by each inequality.

In this exercise, solving both parts individually led us to the same solution: \( x \leq 6 \).

Interval Notation

Interval notation is a concise way to represent a range of values. It uses brackets and parentheses:

• (a, b): Values between a and b, not including a and b.
• [a, b]: Values between a and b, including both a and b.
• (-∞, a): All values less than a.
• (a, ∞): All values greater than a.

In the exercise, the solution \( x \leq 6 \) describes all numbers less than or equal to 6. Using interval notation, we write this as \((-\infty, 6]\). This indicates that every number from -∞ up to 6, including 6, is part of the solution.

Graphing Inequalities

Graphing inequalities helps visualize solutions. By sketching a number line, you can easily see the range of values that fulfill the inequality conditions.
**On a Number Line:**

• Solid Dot: Represents that the number is included in the solution, used for \( ≤ \) or \( ≥ \).
• Open Dot: Represents that the number is not included, used for \( < \) or \( > \).
• Shading: Indicate where all the valid solutions lie on the number line, typically to the left or right of a dot.

For \( x \leq 6 \), place a solid dot on 6 to include it, and shade to the left. This shading shows the viewer that any number to the left of 6 satisfies the inequality.

Algebraic Expressions

Algebraic expressions form the backbone of solving inequalities. These expressions contain variables and constants combined using arithmetic operations. In the inequality \( \frac{7}{3}x + 2 \leq 16 \), the expression \( \frac{7}{3}x + 2 \) combines a fraction with a variable and a constant.
**Breaking Down the Expression:**

• Fraction: Multiplying or dividing both sides helps simplify and isolate \( x \).
• Constant: Used to balance equations or inequalities by adding or subtracting from both sides.

Understanding these components helps solve inequalities systematically. Always simplify expressions to make inequality solving as straightforward as possible.