Problem 88 - Intermediate Algebra

Question

Use a graphing calculator to solve each inequality. Write the solution set using interval notation. See Using Your Calculator: Solving Linear Inequalities in One Variable. $$ 3 x-4 \leq 2 x+4 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 8]\).

Step 1: Understand the Inequality

The given inequality is \( 3x - 4 \leq 2x + 4 \). Our goal is to find the values of \( x \) that make this inequality true. We are looking for these values in interval notation.

Step 2: Simplify the Inequality

Subtract \( 2x \) from both sides to move the \( x \)-terms to one side of the inequality: \( 3x - 4 - 2x \leq 2x + 4 - 2x \). This simplifies to \( x - 4 \leq 4 \).

Step 3: Solve for x

Add 4 to both sides to isolate \( x \): \( x - 4 + 4 \leq 4 + 4 \). This simplifies to \( x \leq 8 \).

Step 4: Write the Solution in Interval Notation

The solution \( x \leq 8 \) means \( x \) can be any value less than or equal to 8. In interval notation, this is \( (-\infty, 8] \).

Step 5: Verify Using a Graphing Calculator

Graph the two equations \( y = 3x - 4 \) and \( y = 2x + 4 \) on a graphing calculator. Identify where \( y_1 \) (\( 3x - 4 \)) is less than or equal to \( y_2 \) (\( 2x + 4 \)). The correct region should be when \( x \leq 8 \).

Key Concepts

Interval Notation

When discussing solutions to inequalities, representing the set of all possible solutions in a concise manner is vital. This is where interval notation comes into play. Interval notation provides a way to describe the range of numbers involved in the solution.

• "(" or ")" (curved brackets) denote that an endpoint is not included in the interval.
• "[" or "]" (square brackets) indicate that an endpoint is included.

For the given solution, the inequality is written as \( x \leq 8 \). In interval notation, this is expressed as \((-\infty, 8]\). Here, the parentheses around \(-\infty\) indicate that it is not a boundary that x can actually reach—it stretches infinitely. The square bracket around 8, however, shows that 8 is a included in the solution set.Interval notation is an efficient way to represent sets, especially when dealing with inequalities that have infinite solutions to one or more sides.

Linear Inequalities

Linear inequalities involve expressions where variables are multiplied by constants and added to or subtracted from each other, usually expressed like \( ax + b \leq cx + d \). These inequalities are similar to linear equations, but instead of an "equals" sign, they use inequality symbols such as \( <, >, \leq, \) and \( \geq \).Linear inequalities can be visualized on a number line or a coordinate plane. They help us determine a range of values for which the inequality holds true. In practical terms, solving linear inequalities involves:- Rearranging terms to isolate the variable on one side.- Simplifying the expression by combining like terms when possible.- Using methods such as addition, subtraction, multiplication, or division to solve for the unknown.For example, with the inequality \( 3x - 4 \leq 2x + 4 \), the goal is to rearrange and simplify until you find \( x \leq 8 \). This signifies that, graphically, the line representing \( 3x - 4 \) lies below or on top of the line for \( 2x + 4 \), whenever x is 8 or less.

Solving Inequalities

Solving inequalities is akin to solving equations, yet requires careful attention to the direction of the inequality sign. The main goal is to isolate the variable on one side to determine which values satisfy the inequality.Steps to solve inequalities typically include:- Moving terms containing the variable to one side of the inequality and constant terms to the other using addition or subtraction.- Combining like terms if necessary to simplify the inequality.- Applying multiplication or division to both sides to solve for the variable, making sure to reverse the inequality sign when multiplying or dividing by a negative.In our problem: 1. We first subtracted \(2x\) from both sides, simplifying to \(x - 4 \leq 4\).2. Then, by adding 4 to both sides, we isolated x and found \(x \leq 8\).These steps must be calculated while keeping the inequality balanced, ensuring the logical relationship between the two sides remains consistent. Finally, verifying the solution graphically or with a calculator provides confirmation of the solution set, demonstrating where one function remains below or meets the other.