Question

which are correct representations of the inequality (6x>3 + 4(2x - 1))? select three options.
(square1>2x)
(square6x>3 + 8x-4)
(square)
(square)
(square)

Explanation:

Step1: Simplify the inequality

$$\begin{align*} 6x&>3 + 4(2x-1)\\ 6x&>3+8x - 4\\ \end{align*}$$

Step2: Move the terms with \(x\) to one side

$$\begin{align*} 6x-8x&>3 - 4\\ -2x&>-1 \end{align*}$$

Step3: Solve for \(x\)

Divide both sides by \(- 2\). When dividing an inequality by a negative number, the direction of the inequality sign changes.
$$x<\frac{1}{2}$$

Now let's check each option:

• For \(1 > 2x\), solving for \(x\) gives \(x<\frac{1}{2}\) (divide both sides by \(2\)).
• \(6x>3 + 8x-4\) is the result of expanding \(4(2x - 1)\) in the original inequality \(6x>3+4(2x - 1)\).
• The number - line that starts from the left (negative direction) and has an open - circle at \(0.5\) (since \(x < 0.5\)) and goes to the left is the correct graphical representation. The first number - line (the one that goes from the leftmost part of the number - line up to \(0.5\) with an open - circle at \(0.5\)) is correct.

Answer:

The correct options are \(1>2x\), \(6x > 3+8x - 4\), and the first number - line (the one that starts from the left and has an open - circle at \(0.5\) and goes to the left).