Problem 34 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 9 x-(10 x-12)<22 $$

Step-by-Step Solution

Answer

The solution is \((-10, \infty)\) and x > -10.

Step 1: Distribute negative sign

Begin by distributing the negative sign in the expression \( -(10x - 12) \) to get \(-10x + 12\).

Step 2: Simplify the inequality

Rewrite the inequality with the distributed terms: \( 9x - 10x + 12 < 22 \). Simplify the terms to get \( -x + 12 < 22 \).

Step 3: Isolate variable

Subtract 12 from both sides of the inequality to isolate the term with \(x\): \(-x < 10\).

Step 4: Solve for x

To solve for \(x\), multiply both sides by -1, and reverse the inequality sign: \( x > -10 \).

Step 5: Express in Interval Notation

The solution set in interval notation is \((-10, \infty)\).

Step 6: Graph the Solution Set

On a number line, draw an open circle at -10 and shade the line indefinitely to the right to represent all numbers greater than -10.

Key Concepts

Interval Notation

Interval notation is a method of expressing a range of values, often as a solution to an inequality. It uses parentheses and brackets to denote open and closed ends of an interval. For example, if a number is greater than (-10 but does not include -10 itself, the corresponding interval notation is \((-10, \infty)\). This means that the numbers we are considering go from just above (-10 to positive infinity.

• Parentheses \(()\) indicate that the endpoint is not included in the interval, known as an open interval.
• Brackets \([]\) indicate that the endpoint is included, known as a closed interval.

Hence, in our exercise, since \(x\) must be greater than (-10, (-10 is not part of the solution set, so we use an open parenthesis. Infinity is always represented with a parenthesis since it is not a tangible number.

Graphing Inequalities

Graphing inequalities is a visual way to represent the solutions on a number line. It helps in understanding the solution set by showing which parts of the number line are included in the inequality solution.

To graph the solution \(x > -10\):

• Start by drawing a number line.
• Place an open circle on (-10 to show it is not included in the solution.
• Shade the line indefinitely to the right of (-10 to represent all the numbers greater than (-10.

This clear visual demonstrates that the solution includes every number to the right of (-10, indicating all possible values of x that satisfy the inequality.

Solving Inequalities

Solving inequalities involves finding the set of values for the variable that makes the inequality true. Much like solving equations, it requires operations such as addition, subtraction, multiplication, or division.

However, when multiplying or dividing both sides of an inequality by a negative number, remember to flip the inequality sign.

The solution of our exercise follows these steps:

• Distribute the negative sign to simplify the expression \(-(10x - 12)\) to \(-10x + 12\).
• Combine like terms to simplify further to \(-x + 12 < 22\).
• Isolate the variable term by subtracting 12 from both sides, yielding \(-x < 10\).
• Finally, solve for \(x\) by multiplying by -1 and flipping the inequality sign: \(x > -10\).

This process helps resolve inequalities step by step ensuring no errors in signs and isolating the variable correctly.

Distributive Property

The distributive property is a fundamental algebraic property used to multiply a single term by two or more terms inside parentheses. It is expressed as: \(a(b + c) = ab + ac\).

In cases where there is a negative sign, distribute it to each term inside the parentheses. For example, in our exercise, \(-(10x - 12)\) becomes \(-10x + 12\) when the negative sign is multiplied through.

Using the distributive property simplifies the expressions so that they are easier to manage, whether you are expanding them or factoring them back down.

• It helps eliminate parentheses, creating a clear equation or inequality.
• Simplifies calculations by breaking down complex expressions into simpler parts.

The distributive property is particularly crucial in solving inequalities, as it ensures all parts of an equation are correctly expanded.