Question

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

Explanation:

Step1: Expand the right - hand side

First, expand $4(2x - 1)$ using the distributive property $a(b - c)=ab - ac$. Here, $a = 4$, $b = 2x$, and $c = 1$. So $4(2x-1)=8x - 4$. The original inequality $6x\geq3 + 4(2x - 1)$ becomes $6x\geq3+8x - 4$.

Step2: Simplify the right - hand side

Combine like terms on the right - hand side: $3+8x - 4=8x - 1$. So the inequality is $6x\geq8x - 1$.

Step3: Isolate the variable terms

Subtract $8x$ from both sides of the inequality: $6x-8x\geq8x - 1-8x$. This gives $-2x\geq - 1$.

Step4: Solve for $x$

Divide both sides of the inequality $-2x\geq - 1$ by $-2$. When dividing an inequality by a negative number, the direction of the inequality sign changes. So $x\leq\frac{1}{2}=0.5$.

Step5: Analyze each option

• Option 1: Starting from $6x\geq8x - 1$, subtract $6x$ from both sides and add 1 to both sides: $1\geq2x$, which is equivalent to $x\leq0.5$. This is correct.
• Option 2: $6x\geq3 + 8x - 4$ is the result of expanding $4(2x - 1)$ in the original inequality, so this is correct.
• Option 3: The number - line represents $x\leq0.5$ (a closed circle at $0.5$ and the line going to the left), so this is correct.
• Option 4: The number - line represents $x\geq0.5$, which is incorrect.
• Option 5: The number - line represents $x\geq0.5$, which is incorrect.

Answer:

1 ≥ 2x, 6x ≥ 3 + 8x - 4, the number - line with a closed circle at 0.5 and the line going to the left (the first number - line)