Problem 50 - College Algebra

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(6(x-1)-(4-x) \geq 7 x-8\)

Step-by-Step Solution

Answer

The solution set is \(\emptyset\) represented by an open interval with no values, and there are no values of 'x' on the number line.

Step 1: Simplify both sides

Expand the terms on the left side to get: \(6x - 6 - 4 + x \geq 7x - 8\). Combine like terms to simplify it further to: \(7x - 10 \geq 7x - 8\).

Step 2: Rearrange to isolate x

Subtract \(7x\) from both sides of the inequality to get: \(-10 \geq -8\). This yields a false statement, implying there is no 'x' that can satisfy this inequality.

Step 3: Write solution set in interval notation and graph

Since there is no 'x' that can satisfy the inequality, the solution set is the empty set, represented as \(\emptyset\). In terms of interval notation, it is denoted as \(\) (an empty interval). The graph on the number line will also show no value for 'x'.

Key Concepts

Understanding Linear Inequalities

Linear inequalities are very similar to linear equations. While equations show equality, inequalities show a relationship where one side can be greater or lesser than the other. In mathematics, this is expressed using symbols like ">" (greater than), "<" (less than), "≥" (greater than or equal to), and "≤" (less than or equal to).

Solving a linear inequality normally involves simplifying expressions on either side and trying to isolate the variable. Similar to equations, you can add, subtract, multiply, or divide on both sides. However, a special rule applies: when multiplying or dividing both sides by a negative number, the inequality sign must be reversed.

In the given exercise, the inequality is expressed with "≥", leading us to look for all possible values of 'x' that satisfy the inequality or demonstrate it as false.

Defining the Solution Set

The solution set of an inequality comprises all possible values of the variable that make the inequality true. If a solution is discovered, it's denoted using interval notation or a listing of numbers.

Interval notation involves the use of brackets and parentheses to show sets of numbers that satisfy the inequality. For example, if an inequality is true for 'x' values greater than 3 and less than 10, the solution set in interval notation would be written as (3, 10).

In the exercise provided, the simplification of the inequality showcased a contradiction: - When rearranged, -10 is not greater than or equal to -8. There's no solution for 'x', hence it results in an empty set, denoted as \(\emptyset\) in interval notation.

Role of the Number Line

The number line is a crucial tool for visually representing solutions of inequalities. It helps in understanding which portions of the line are occupied by potential solutions.

When graphing an inequality solution set on a number line:

• Closed circles are used to include boundary values (when inequality is ">=" or "<=").
• Open circles exclude boundary values (for ">" or "<").
• Arrows indicate that numbers continue indefinitely in a direction.

In the exercise's practical application, since there are no values for 'x' fulfilling the inequality, the number line remains blank for this case, illustrating an empty set.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key step in solving inequalities. It involves:

• Removing parentheses by distributing multiplication over addition/subtraction.
• Combining like terms — those with the same variables raised to the same power.

For the given exercise:

• The expression "\(6(x-1)-(4-x)\)" is expanded using distribution.
• Simplifying leads to combining like terms, yielding "\(7x - 10 \geq 7x - 8\)."

By refining expressions to their simplest form, you're a step closer to isolating variables and solving the inequality. Understanding these processes lays a foundation essential for tackling more complex algebraic problems.