Problem 57 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. (lesson \(1-5 )\) $$ 7 x+11 > 9 x+3 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 4\), graph it with an open circle at 4 and shade left.

Step 1: Begin Isolating the Variable

Start by moving all terms involving \(x\) to one side of the inequality. To begin, subtract \(7x\) from both sides of the inequality: \[ 7x + 11 - 7x > 9x + 3 - 7x \] which simplifies to \[ 11 > 2x + 3 \].

Step 2: Simplify the Inequality

Now, subtract 3 from both sides to further isolate the term with \(x\): \[ 11 - 3 > 2x + 3 - 3 \] which simplifies to \[ 8 > 2x \].

Step 3: Solve for the Variable

Divide both sides by 2 to solve for \(x\): \[ \frac{8}{2} > \frac{2x}{2} \] which simplifies to \[ 4 > x \] or equivalently \( x < 4 \).

Step 4: Graph the Solution

On a number line, draw an open circle at 4 to indicate that 4 is not included in the solution set. Shade the region to the left of 4 to represent all numbers less than 4 that satisfy the inequality.

Key Concepts

Algebraic Expressions

Algebraic expressions are at the heart of solving inequalities like the one given in the exercise. An algebraic expression is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, or division. In the given inequality, \(7x + 11 > 9x + 3\), both sides of the inequality are algebraic expressions.

Understanding how to manipulate these expressions is crucial. By rearranging terms and simplifying expressions, we can isolate the variable, making it easier to solve the inequality. Always remember, the equality and inequality rules work similarly when combining like terms or rearranging expressions. To maintain inequality, the operations must be applied consistently on both sides. This step allows us to simplify the problem, making it more manageable and set the stage for solving it.
Approaching algebraic expressions step by step, checking every calculation, and ensuring each transformation is valid is pivotal to getting the correct solution.

Variable Isolation

Variable isolation is a fundamental skill in solving inequalities, similar to solving equations. The goal is to manipulate the expression such that the variable stands alone on one side of the inequality sign. This process involves applying inverse operations to "move" the variable and simplify the inequality. In the solution provided, variable isolation involved several steps.

• First, the terms involving the variable \(x\) were placed on one side by subtracting \(7x\) from both sides, simplifying the inequality to \(11 > 2x + 3\).
• Then, constant terms were isolated by subtracting 3 from both sides, resulting in \(8 > 2x\).
• Finally, \(x\) was isolated by dividing both sides by 2, leading to \(x < 4\).

Each of these steps involves applying basic arithmetic operations equally to both sides of the inequality, keeping the inequality balanced. Variable isolation is a methodical process and crucial for obtaining a readable and interpretable solution.

Inequality Graphing

Graphing inequalities is a visual way to represent the solution set. After finding the inequality \(x < 4\), graphing helps us understand what this means in terms of values \(x\) can take. In this graphing process, different symbols and shading apply compared to equations.

• First, you place an open circle on the number 4 on a number line. The open circle indicates that 4 is not included in the solution set.
• Then, shade the area to the left of the number 4. This shaded region represents all possible solution values that satisfy the inequality, meaning \(x\) can take any value less than 4.

This graphical representation provides an intuitive understanding of all possible values \(x\) can have, making it easier to visualize and interpret the inequality solution.

Number Line

A number line is a valuable tool in solving and graphing inequalities. It stretches horizontally and allows us to visually place numbers and intervals.

When solving inequalities, the number line helps us determine which numbers satisfy the inequality conditions. Using the example from the problem \(x < 4\), we use a number line to visually depict values that are less than 4.

Use these easy steps:

• Draw a horizontal line and mark even numerical intervals on this line, ensuring clarity in the representation.
• Use an open circle on the number 4, which indicates that the number itself is not included as a solution.
• Shade the part of the line that extends to the left of 4, marking all numbers that solve \(x < 4\).

This clear visual marker helps identify and understand where the solutions to the inequality exist on the real number line.