Question

which are correct representations of the inequality ( 6x geq 3 + 4(2x - 1) )? select three options.
( square 1 geq 2x )
( square 6x geq 3 + 8x - 4 )
( square ) number line: arrow left from 0.5, marked at -1.5, -1, -0.5, 0, 0.5, 1, 1.5
( square ) number line: arrow right from -0.5, marked at -1.5, -1, -0.5, 0, 0.5, 1, 1.5
( square ) number line: arrow right from 0.5, marked at -1.5, -1, -0.5, 0, 0.5, 1, 1.5

Explanation:

Step1: Simplify the right - hand side of the inequality

First, we expand the expression \(4(2x - 1)\) using the distributive property \(a(b + c)=ab+ac\). Here \(a = 4\), \(b=2x\) and \(c=- 1\), so \(4(2x-1)=4\times2x-4\times1 = 8x - 4\).
The original inequality is \(6x\geq3 + 4(2x - 1)\), after expanding the right - hand side, we get \(6x\geq3 + 8x-4\). So the second option \(6x\geq3 + 8x - 4\) is correct.

Step2: Solve the inequality \(6x\geq3 + 8x-4\)

First, simplify the right - hand side: \(3-4=-1\), so the inequality becomes \(6x\geq8x - 1\).
Then, subtract \(6x\) from both sides: \(6x-6x\geq8x - 6x-1\), which simplifies to \(0\geq2x - 1\).
Next, add 1 to both sides: \(0 + 1\geq2x-1 + 1\), so \(1\geq2x\) (or \(2x\leq1\), which is equivalent to \(x\leq\frac{1}{2}=0.5\)). So the first option \(1\geq2x\) is correct.

Step3: Analyze the number line representations

We have the solution \(x\leq0.5\). On a number line, this means we have a closed circle at \(x = 0.5\) (because the inequality is non - strict, \(\leq\)) and the arrow points to the left (towards negative infinity) to represent all values of \(x\) less than or equal to \(0.5\).
Looking at the number line options:

• The first number line (with the arrow pointing left and a closed circle at \(0.5\)) represents \(x\leq0.5\).
• The second number line has an arrow pointing right, which would represent \(x\geq - 0.5\), which is incorrect.
• The third number line has a closed circle at \(0.5\) and an arrow pointing right, which is incorrect. Wait, no, wait. Wait, the first number line (the one with the arrow to the left, closed circle at \(0.5\)): let's re - check. The solution is \(x\leq0.5\). So the number line with a closed dot at \(0.5\) and the arrow going to the left (towards \(-\infty\)) is correct. Also, when we solve \(1\geq2x\), dividing both sides by 2 (since 2>0, the inequality sign remains the same) gives \(x\leq0.5\). So the number line with the closed circle at \(0.5\) and the arrow to the left (the first number line among the three number line options) and the first two (the \(1\geq2x\) and \(6x\geq3 + 8x - 4\)) and the correct number line. Wait, the three correct options are:

1. \(1\geq2x\)
2. \(6x\geq3 + 8x - 4\)
3. The number line with a closed circle at \(0.5\) and the arrow pointing to the left (the first number line option among the three number line choices)

Answer:

• \(1\geq2x\)
• \(6x\geq3 + 8x - 4\)
• The number line with a closed circle at \(0.5\) and the arrow pointing to the left (the first number line option in the given set of number lines)