Question

solve the inequality and graph the solution. 6x - 7x ≥ 4x + 3. choose the correct graph below. (type an inequality. use integers or fractions for any numbers in the inequality.) the solution is

Explanation:

Step1: Simplify the inequality

Start with \(6x - 7x\geq4x + 3\). Combine like - terms on the left - hand side: \((6x-7x)=-x\), so the inequality becomes \(-x\geq4x + 3\).

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

Add \(x\) to both sides of the inequality: \(-x+x\geq4x + x+3\), which simplifies to \(0\geq5x + 3\). Then subtract 3 from both sides: \(0 - 3\geq5x+3 - 3\), resulting in \(-3\geq5x\).

Step3: Solve for \(x\)

Divide both sides of the inequality \(-3\geq5x\) by 5. Since we are dividing by a positive number, the direction of the inequality sign remains the same. So \(x\leq-\frac{3}{5}\).

Step4: Analyze the graph

The solution \(x\leq-\frac{3}{5}\) is represented on a number line with a closed circle at \(-\frac{3}{5}\) (if the inequality was \(\leq\) or \(\geq\)) or an open circle at \(-\frac{3}{5}\) (if the inequality was \(<\) or \(>\)) and an arrow pointing to the left (because \(x\) is less than or equal to \(-\frac{3}{5}\)). In this case, since we have \(x\leq-\frac{3}{5}\), we should have a closed circle at \(-\frac{3}{5}\) and an arrow pointing to the left.

Answer:

The solution is \(x\leq-\frac{3}{5}\). Looking at the graphs, if the graph has a closed circle at \(-\frac{3}{5}\) and an arrow pointing to the left, that is the correct graph. Without seeing the exact details of the circles (open or closed) in your options, but based on the solution, the correct graph should represent all values of \(x\) that are less than or equal to \(-\frac{3}{5}\). If we assume the standard convention of open circle for \(<\) and \(>\) and closed circle for \(\leq\) and \(\geq\), and among the options with an arrow pointing left from \(-\frac{3}{5}\), the one with a closed circle at \(-\frac{3}{5}\) is the correct one.