How do you solve and graph the compound inequality #4v-3<=-27# and #-3v>=-3# ?

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To solve the compound inequality (4v - 3 \leq -27) and (-3v \geq -3), we'll solve each inequality separately and then combine the solutions.

1. Solve (4v - 3 \leq -27): [4v - 3 \leq -27] Add 3 to both sides: [4v \leq -24] Divide both sides by 4: [v \leq -6]

2. Solve (-3v \geq -3): [-3v \geq -3] Divide both sides by -3 (remember to flip the inequality sign when dividing or multiplying by a negative number): [v \leq 1]

Combining the solutions, we get (v \leq -6) and (v \leq 1). Since both inequalities have (\leq), we take the intersection of the solutions, which means we choose the smaller value for (v). Therefore, the solution is (v \leq -6).

To graph the solution on a number line, we'll mark -6 on the left side and shade everything to the left of -6, including -6 itself. This represents all the values of (v) that satisfy the compound inequality.

To solve the compound inequality (4v - 3 \leq -27) and (-3v \geq -3), follow these steps:

For the first inequality:

1. Add 3 to both sides: (4v \leq -24)
2. Divide both sides by 4: (v \leq -6)

For the second inequality:

1. Divide both sides by -3, remembering to reverse the inequality sign: (v \leq 1)

So, the solution to the compound inequality is (v \leq -6) and (v \leq 1).

To graph this compound inequality, plot the solution on a number line, remembering to shade in the regions where the variable satisfies both inequalities, which is where they overlap.