Questions: Express the set -2x + 18 ≥ 4x + 12 using interval notation.

Transcript text: Express the set $-2 x+18 \geq 4 x+12$ using interval notation.

Solution

Solution Steps

To solve the inequality $-2x + 18 \geq 4x + 12$, we need to isolate $x$ on one side of the inequality. This involves moving all terms involving $x$ to one side and constant terms to the other. After simplifying, we solve for $x$ and express the solution in interval notation.

Step 1: Move all terms to one side

We start with the inequality:

\[ -2x + 18 \geq 4x + 12 \]

To simplify, we move all terms involving $x$ to one side and constant terms to the other side. Subtract $4x$ from both sides:

\[ -2x - 4x + 18 \geq 12 \]

This simplifies to:

\[ -6x + 18 \geq 12 \]

Step 2: Isolate the variable term

Next, we subtract 18 from both sides to isolate the term with $x$:

\[ -6x + 18 - 18 \geq 12 - 18 \]

This simplifies to:

\[ -6x \geq -6 \]

Step 3: Solve for $x$

To solve for $x$, divide both sides by $-6$. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:

\[ x \leq 1 \]

Step 4: Express in interval notation

The solution $x \leq 1$ in interval notation is:

\[ (-\infty, 1] \]