Problem 35 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 2(x-7)-3(x+3) \leq-3 $$

Step-by-Step Solution

Answer

The solution set is \(x \geq -20\), interval notation is \([-20, \infty)\).

Step 1: Distribute and Simplify

First, we will distribute the numbers outside the parentheses through the expressions inside the parentheses. This means multiplying both 2 by each term in the parenthesis \((x-7)\) and -3 by each term in \((x+3)\). So we get: \[ 2x - 14 - 3x - 9 \]. Next, combine like terms, which simplifies to \[ -x - 23 \]. Therefore, the inequality becomes: \[ -x - 23 \leq -3 \].

Step 2: Isolate the Variable

To isolate \(x\), first add 23 to both sides of the inequality:\[ -x - 23 + 23 \leq -3 + 23 \], which simplifies to \[ -x \leq 20 \]. Then, multiply both sides by -1 while flipping the inequality sign (since multiplying or dividing by a negative number reverses the inequality): \[ x \geq -20 \].

Step 3: Express in Interval Notation

The solution \(x \geq -20\) means that \(x\) is greater than or equal to -20. In interval notation, this is expressed as: \[ [-20, \infty) \].

Step 4: Graph the Solution on a Number Line

On the number line, represent the solution set with a solid circle at -20, indicating that -20 is included in the solution set. Then draw an arrow extending to the right from -20, indicating that all numbers greater than -20 are included in the solution set.

Key Concepts

Interval Notation

Interval notation is a method used to describe a range of values that belong to a solution set. This is particularly helpful when solving inequalities, as it provides a concise way to express all the possible solutions.
For example, if we say that a number, like \(x\), is greater than or equal to \(-20\), we are talking about all numbers starting from \(-20\) and moving towards positive infinity. In interval notation, this is written as:

• \([-20, \infty)\)

• The bracket "[" at \(-20\) indicates \(-20\) is included in the interval.
• The parenthesis ")" at \(\infty\) suggests infinity is not a reachable number, so it can't be "included" in the interval.

This notation efficiently communicates that \(x\) includes \(-20\) and all numbers after it, infinitely extending.

Distributive Property

The distributive property is a critical algebraic tool that allows us to clear parentheses by distributing a number across the terms inside.
To use the distributive property, multiply the term outside the parentheses by each term inside. Let's apply this idea to the expression given:

• Consider the expression: \(2(x-7)\)
• Distribute the 2 to both terms inside the parentheses: \(2 \cdot x\) and \(2 \cdot -7\)
• This gives us \(2x - 14\)
• Do the same for \(-3(x+3)\), giving \(-3x - 9\)

Putting these results together, you have: \(2x - 14 - 3x - 9\).
Utilizing the distributive property correctly ensures that expressions are simplified accurately, making it easier to solve the inequality as seen in the steps.

Graphing Solution Sets

Graphing a solution set on a number line provides a visual representation of all values that satisfy the inequality. This visual can make it easier to understand and communicate the solution effectively.
For the inequality solution \(x \geq -20\):

• Place a solid circle on the number \(-20\) on the number line. The solid circle signifies that \(-20\) is included in the solution set.
• Draw a line or arrow extending to the right from \(-20\), indicating that all numbers greater than \(-20\) are a part of the solution set.

This graphically demonstrates that the solution set includes all numbers from \(-20\) onward. Such graphical representation clarifies the relationship defined by the inequality and can be a useful tool for visual learners.

Algebraic Manipulation

Algebraic manipulation involves using algebraic methods to rearrange and simplify expressions to isolate and solve for variables.
In solving the inequality \(-x - 23 \leq -3\), follow these steps to isolate \(x\):

• Add 23 to both sides: \(-x - 23 + 23 \leq -3 + 23\), resulting in \(-x \leq 20\).
• Since \(-x\) is negative, multiply both sides by \(-1\) to make \(x\) positive. Remember to flip the inequality sign: \(x \geq -20\).

This manipulation is a fundamental part of algebra that allows us to transform an inequality into an understandable solution.
By understanding and applying these steps, we are better able to solve inequalities efficiently, arriving at a solution that can be expressed using interval notation and graphing. This is a vital skill in algebra that supports various applications in mathematics.