Question

which number line represents the solution set for the inequality ( 2x - 8 geq 5x - 2 + 3 )?

Explanation:

Step1: Simplify the right - hand side of the inequality

First, simplify \(5(x - 2)+3\).
Using the distributive property \(a(b + c)=ab+ac\), we have \(5(x - 2)+3=5x-10 + 3=5x-7\).
The inequality becomes \(2x-8>5x - 7\).

Step2: Move the \(x\) terms to one side and the constants to the other side

Subtract \(2x\) from both sides: \(2x-8-2x>5x - 7-2x\), which simplifies to \(-8>3x - 7\).
Then add \(7\) to both sides: \(-8 + 7>3x-7 + 7\), so \(-1>3x\).

Step3: Solve for \(x\)

Divide both sides by \(3\): \(\frac{-1}{3}>x\), or \(x<-\frac{1}{3}\approx - 0.33\).

Answer:

The number line that has an open circle (since the inequality is strict, \(>\) which corresponds to \(<\) after rearrangement) and the arrow pointing to the left (towards smaller numbers) starting from approximately \(-0.33\) is the correct one. Looking at the options (assuming the first option has the arrow pointing left starting from around \(-0.33\)), that is the number line representing the solution set of the inequality.