Problem 119 - Precalculus Essentials

Question

Solve and graph the solution set on a number line. $$\frac{x+3}{4} \geq \frac{x-2}{3}+1$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq 5\). This is represented on a number line with a filled-in circle at 5 and a line extending to the left.

Step 1: Simplify the Inequality

Algebraically simplify the inequality. Start by multiplying both sides by 12 to eliminate the fractions: \(12*\frac{x+3}{4} \geq 12*\frac{x-2}{3} + 12*1\) which simplifies to \(3(x+3) \geq 4(x-2) + 12\).

Step 2: Solve for x

Distribute and combine like terms to isolate x on one side of the inequality. This yields: \(3x+9 \geq 4x-8+12\) which simplifies further to \(3x+9 \geq 4x+4\). Solving for x, this results in \(x \leq 5\).

Step 3: Graph the Solution Set on a Number Line

Draw a number line and mark the number 5. As the solution to the inequality is \(x \leq 5\), fill in the circle at 5 and draw a line extending to the left to show that all numbers that are less than or equal to 5 are part of the solution set.

Key Concepts

Algebraic Simplification

Algebraic simplification is a crucial step in solving inequalities as it helps to eliminate fractions, combine like terms, and ultimately isolate the variable. This makes the inequality easier to manage. To simplify the given inequality \(\frac{x+3}{4} \geq \frac{x-2}{3}+1\), the fractions need to be removed first. This is done by finding a common multiple—12 in this case, and multiplying every term by it.

• This starts as: \(12*\frac{x+3}{4} \geq 12*\frac{x-2}{3} + 12*1\).
• The result is \(3(x+3) \geq 4(x-2) + 12\).

Next, distribute the numbers across the parentheses to simplify further. That becomes \(3x + 9 \geq 4x - 8 + 12\). After performing these arithmetic operations, combining like terms will continue the simplification; here, it’s \(3x + 9 \geq 4x + 4\). Rearranging this inequality eventually leads to isolating \(x\), providing the solution \(x \leq 5\). Using algebraic simplification not only cleans the inequality but also paves the way for solving it efficiently.

Graphing Inequalities

Graphing inequalities visually represents the solution set on a coordinate plane or a number line, making them more intuitive to understand. For this exercise, once the inequality is simplified to \(x \leq 5\), graphing can illustrate all the potential solutions. When graphing inequalities, you need to consider the type of inequality sign to determine how the solution is represented:

• \(x \leq 5\) means that the number 5 is included in the solution set, signified by a closed or filled circle.
• If the inequality were strict, like \(x < 5\), an open circle would be used instead.

On the number line, after marking the point 5, draw a line extending to the left to represent all numbers less than or equal to 5. This approach assists in understanding and verifying the solution, ensuring all conditions of the inequality are met visually.

Number Line

A number line is a simple but effective tool to depict the range of solutions for inequalities, fostering better comprehension of numerical relationships. When solving an inequality and expressing its solutions, the number line offers a visual cue of where these solutions lie. For the inequality \(x \leq 5\), the number line will illustrate this as follows:- A point at 5 which is filled or closed, showing that 5 itself is part of the solution (inclusive).- A shaded area extending to the left from 5, visually highlighting that all numbers less than 5 are also valid solutions.The number line simplifies the complex idea of infinite solutions by presenting them in a straightforward, clear format. Thus, it not only supports problem-solving but also strengthens intuitive learning of mathematical concepts. Using such a visual representation can be a powerful method to grasp and verify solutions effectively.