Solve the Inequality for x 3x+8> =4x

Solution:

Step 1: Isolate the variable $x$ on one side of the inequality.

• Step 1.1: Subtract $4x$ from both sides to move the $x$ terms to one side.

  $3x + 8 - 4x \geq 4x - 4x$

• Step 1.2: Combine like terms on the left side.

  $-x + 8 \geq 0$

Step 2: Eliminate the constant term from the side with the variable.

• Subtract $8$ from both sides to isolate the $x$ term.

  $-x \geq -8$

Step 3: Solve for $x$ by dividing by the coefficient of $x$.

• Step 3.1: Divide by $-1$, remembering to reverse the inequality sign.

  $\frac{-x}{-1} \leq \frac{-8}{-1}$

• Step 3.2: Simplify the inequality.

  • Step 3.2.1: A negative divided by a negative is a positive.

    $\frac{x}{1} \leq \frac{-8}{-1}$

  • Step 3.2.2: Simplify the division by $1$.

    $x \leq \frac{-8}{-1}$

Step 4: Finalize the solution.

• Step 3.3: Simplify the right side of the inequality.

  $x \leq 8$

• Present the solution in different notations.

  Inequality Form: $x \leq 8$ Interval Notation: $(-\infty, 8]$

Knowledge Notes:

To solve an inequality, similar to solving an equation, you need to isolate the variable on one side. However, there are some key differences to keep in mind:

1. Combining Like Terms: When you have the same variable on both sides of the inequality, you'll want to combine them by adding or subtracting terms to get the variable on one side only.

2. Reversing the Inequality: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is because multiplying or dividing by a negative number changes the order of the numbers on the number line.

3. Division by Negative One: In this specific problem, we divide by $-1$ to solve for $x$. This action requires us to flip the inequality sign.

4. Interval Notation: The solution to an inequality can be expressed in interval notation, which provides a way to denote the set of all possible solutions. In this case, $(-\infty, 8]$ means all numbers less than or equal to 8.

5. Inequality Signs: The inequality sign $\leq$ means "less than or equal to," while $\geq$ means "greater than or equal to." It is crucial to maintain the correct direction of these signs throughout the problem-solving process.