Problem 84 - 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) \(6+3(1-x) \geq 0\) (b) \(6+3(1-x)<0\)

Step-by-Step Solution

Answer

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

Step 1: Distribute and Simplify the Expression

For inequality (a), start by distributing the 3 in the expression. Rewrite the inequality: \(6 + 3(1-x) \geq 0\) becomes \(6 + 3 - 3x \geq 0\). Simplify it further to get: \(9 - 3x \geq 0\).

Step 2: Isolate the Variable

To solve for \(x\), begin by isolating the \(-3x\). Subtract 9 from both sides: \(-3x \geq -9\). Then, divide each side by \(-3\) (remember, dividing by a negative flips the inequality): \(x \leq 3\).

Step 3: Interval Notation for (a)

Express the solution in interval notation. For \(x \leq 3\), the interval notation is \((-\infty, 3]\).

Step 4: Graphical Representation of Part (a)

Sketch a number line and shade to the left of 3, including 3, to represent the solution \((-\infty, 3]\).

Step 5: Apply Hint for (b)

Use the result from part (a) for part (b). The inequality is the opposite: \(6 + 3(1-x) < 0\) is the negation of \(6 + 3(1-x) \geq 0\).

Step 6: Solution for Part (b)

The solution for \(6 + 3(1-x) < 0\) is the interval that doesn't include the solution from (a). This means \(x > 3\), thus the interval is \((3, \infty)\).

Step 7: Graphical Representation of Part (b)

On the number line, shade to the right of 3, not including 3, to represent the solution \((3, \infty)\).

Key Concepts

Interval Notation

Interval notation is a helpful method for writing the solution of an inequality. It helps to clearly state which numbers are included in the solution set. Here's how interval notation works:

• Use brackets \'[\' or \'\' to indicate that an endpoint number is included in the set (a closed interval).
• Use parentheses \((\) or \()\) to signify that a number is not part of the interval (an open interval).

In our problem, for part (a), the solution is given by the inequality \(x \leq 3\). This means that 3 is included in the solution, so we use a bracket:

• Interval notation: \(( -\infty, 3 ]\).

The use of \(-\infty\) suggests that the solution extends infinitely in the negative direction, encompassing all numbers less than or equal to 3.
For part (b), the inequality is \(x > 3\). Here, 3 is not included, so a parenthesis is used:

• Interval notation: \((3, \infty)\).

This indicates that the solution consists of all numbers greater than 3. Practicing interval notation helps in efficiently communicating solutions in mathematics.

Graphical Representation

Graphical representation is a visual way to display solutions to inequalities on a number line. This method enhances understanding by showing which numbers satisfy the inequality. Here are some basic steps:

• Draw a horizontal line to represent all real numbers.
• Mark the critical point (or points) involved in the inequality. The critical point is where the inequality changes direction or where the boundary condition is met.
• Use a dot or circle at the critical point, a filled circle means the point is included, whereas an open circle shows it's not.

For inequality (a), \(x \leq 3\):

• Draw a solid dot at 3.
• Shade all numbers to the left to indicate \(( -\infty , 3 ]\).

For (b), \(x > 3\):

• Draw an open circle at 3 (since 3 is not included).
• Shade to the right, representing \((3, \infty)\).

Using graphs can transform abstract algebraic solutions into something more interpretable, making it easier for students to grasp the concept of solutions to inequalities.

Algebraic Manipulation

Algebraic manipulation involves transforming expressions and equations to arrive at a solution. It includes operations such as distribution, combining like terms, and isolating variables. This practice is essential in solving inequalities.In step-by-step solutions for inequalities like our example, the main points include:

• Distribute: Apply distribution to eliminate parentheses. For example, in \(6 + 3(1-x)\), distribute 3 across \((1-x)\) to achieve \(6 + 3 - 3x\).
• Simplify: Combine similar terms such as constants or variable terms. This step simplifies the inequality to a point where solving becomes straightforward (\(9 - 3x \geq 0\)).
• Isolate the Variable: Reorganize the expression to get the variable (here, \(x\)) on one side. This involves operations like subtraction or division. Care especially when dividing by a negative number, as it reverses the inequality sign, leading to \(x \leq 3\).

Understanding each operation in detail allows for accurate solving and comprehensive comprehension of inequalities. By systematically applying these techniques, even complex inequalities can be managed effectively.