Problem 54 - Algebra 2

Question

Use a graphing calculator to solve each inequality. \(3(x+3) \geq 2(x+4)\)

Step-by-Step Solution

Answer

The solution is \(x \geq -1\).

Step 1: Simplify the Inequality

First, we'll expand and simplify both sides of the inequality. Distribute the 3 on the left side: \(3(x+3) = 3x + 9\). On the right side, distribute the 2: \(2(x+4) = 2x + 8\). So, the inequality becomes \(3x + 9 \geq 2x + 8\).

Step 2: Isolate the Variable

Next, we need to isolate \(x\) on one side of the inequality. Subtract \(2x\) from both sides: \(3x + 9 - 2x \geq 8\), which simplifies to \(x + 9 \geq 8\).

Step 3: Solve for x

Subtract 9 from both sides to solve for \(x\): \(x + 9 - 9 \geq 8 - 9\). This simplifies to \(x \geq -1\).

Step 4: Graph the Solution

Using a graphing calculator, plot the equation \(3(x+3) - 2(x+4) = 0\). The solution \(x \geq -1\) is visualized as the region where the graph satisfies the inequality, which is where the line is above or on the x-axis for \(x \geq -1\).

Key Concepts

Graphing Calculator

A graphing calculator is an incredibly useful tool for visualizing mathematical problems, such as linear inequalities. It allows you to plot the functions and see where they meet or cross the axes. In our inequality problem, we want to solve \(3(x+3) \geq 2(x+4)\). By using a graphing calculator, you can input the equation in a simplified form as \(3x + 9 - (2x + 8)\) and graph it. This results in the line \(x + 1 = 0\).

• The graph helps you see the areas where inequality conditions hold true.
• You can zoom or adjust the view to get a more accurate picture of your solution.
• This visual aid confirms the algebraic solution.

For this example, graphing confirms that the solution is \(x \geq -1\), by showing the shaded area to the right of \(x = -1\).Use this feature to check your work and understand the solution better.

Linear Inequalities

Linear inequalities are similar to linear equations but involve inequality symbols like \(\geq, \leq, >,\) and \(<\) instead of an equal sign. They represent ranges of values that satisfy the inequality condition, rather than a single point.- For example, the original problem \(3(x+3) \geq 2(x+4)\) compares two linear expressions.- Solving a linear inequality often involves simplifying both sides and isolating the variable, just like solving equations.Linear inequalities can represent real-world constraints, such as budgets or physical limits, where values can vary within certain bounds. Understanding these expressions is crucial for real-world problem-solving.

Solving Inequalities

Solving inequalities involves manipulating the inequality much like an equation, but you must always be cautious with the inequality sign.- Here, we simplified \(3(x+3) \geq 2(x+4)\) to \(x \geq -1\) by isolating \(x\).- Remember, whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.After the algebraic manipulation, verification by graphing or substituting values can help confirm the solution's accuracy. By solving the inequality step-by-step, you gradually narrow down the range of possible solutions satisfying the original problem.

Algebraic Manipulation

Algebraic manipulation is a key process in solving inequalities.

• Start by expanding expressions, as seen with \(3(x+3)\), which becomes \(3x + 9\).
• This is often followed by combining like terms on both sides of the inequality.
• Next, isolate the variable by adding or subtracting terms, as demonstrated with \(x + 9 - 9 \geq 8 - 9\).

These steps help transform complex inequalities into simpler forms, allowing easy determination of the solution set. Always keep track of your steps, especially when adding or subtracting terms, to prevent errors. Understanding how each piece fits together lays a strong foundation for more complex mathematics.