Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.

Question

11xminus9less than or equals9plus5x

Answer 1

To solve the inequality \( 11x - 9 \leq 9 + 5x \) algebraically, we will follow these steps:

1. Rearrange the inequality: Subtract \( 5x \) from both sides: \[ 11x - 5x - 9 \leq 9 \] This simplifies to: \[ 6x - 9 \leq 9 \]

2. Isolate the variable: Add \( 9 \) to both sides: \[ 6x \leq 18 \]

   Now, divide both sides by \( 6 \): \[ x \leq 3 \]

Thus, the solution to the inequality is \( x \leq 3 \).

Graphical Solution

To represent this solution graphically on a number line:

• Draw a horizontal number line.
• Mark the point \( 3 \) on the number line.
• Since the inequality is inclusive (the \( \leq \) symbol), shade the region to the left of \( 3 \) and include the point \( 3 \) with a closed dot.

Interval Notation

The interval notation for the solution \( x \leq 3 \) is: \[ (-\infty, 3] \]

Summary

• Algebraic Solution: \( x \leq 3 \)
• Graphical Solution: The number line is shaded to the left of \( 3 \), including \( 3 \).
• Interval Notation: \( (-\infty, 3] \)