To solve the inequality \(-3x \geq 12\), Mary Ann would follow these steps:
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Isolate the variable: The first step in solving an inequality is to isolate the variable \(x\). In this case, Mary Ann would want to get \(x\) by itself on one side of the inequality.
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Divide both sides by -3: Since the coefficient of \(x\) is \(-3\), she would divide every term of the inequality by \(-3\) to solve for \(x\): \[ -3x \geq 12 \implies x \leq \frac{12}{-3} \] Note: When dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be flipped. Therefore, the inequality changes from \(\geq\) to \(\leq\).
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Simplify: After performing the division, Mary Ann would simplify the right side: \[ x \leq -4 \]
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Write the solution set: The solution set can be expressed in interval notation. The solution \(x \leq -4\) can be written as: \[ (-\infty, -4] \]
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Check the solution: It is also a good practice to check the solution by substituting a value that is less than or equal to \(-4\) back into the original inequality:
- For example, if \(x = -4\): \[ -3(-4) = 12 \quad \text{(which satisfies the inequality)} \]
- If \(x = -5\): \[ -3(-5) = 15 \quad \text{(which also satisfies the inequality)} \] This confirms that the steps taken to solve the inequality were correct.
In summary, the steps Mary Ann took to find the solution set are:
- Isolated \(x\) by dividing both sides by \(-3\),
- Flipped the inequality sign when dividing by a negative,
- Found the simplified result \(x \leq -4\),
- Expressed the solution in interval notation as \((-\infty, -4]\),
- Checked her work by substituting values into the original inequality.
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