which are correct representations of the inequality 6x ≥ 3 + 4(2x - 1)? select three options.\n1 ≥ 2x\n6x ≥…

which are correct representations of the inequality 6x ≥ 3 + 4(2x - 1)? select three options.\n1 ≥ 2x\n6x ≥ 3 + 8x - 4

which are correct representations of the inequality 6x ≥ 3 + 4(2x - 1)? select three options.\n1 ≥ 2x\n6x ≥ 3 + 8x - 4

Answer

Answer:

  1. $1\geq2x$
  2. $6x\geq3 + 8x - 4$
  3. The number - line with a closed - circle at $0.5$ and arrow pointing to the left

Explanation:

Step1: Expand the right - hand side

Expand $4(2x - 1)$ using the distributive property $a(b - c)=ab - ac$. Here $a = 4$, $b = 2x$, $c = 1$, 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: $3-4=-1$, so the inequality is $6x\geq8x - 1$.

Step3: Solve for $x$

Subtract $8x$ from both sides: $6x-8x\geq8x - 1-8x$, which simplifies to $-2x\geq - 1$. Divide both sides by $-2$. When dividing an inequality by a negative number, the direction of the inequality sign changes. So we get $x\leq0.5$, which is equivalent to $1\geq2x$.

Step4: Analyze the number - line

The solution $x\leq0.5$ is represented on a number - line as a closed - circle at $0.5$ (because $x$ can equal $0.5$) and an arrow pointing to the left (since $x$ is less than or equal to $0.5$).