Questions: Solve: 2x+3 ≥ 2

Transcript text: Solve: $|2 x+3| \geq 2$

Solution

Solution Steps

To solve the inequality $ |2x + 3| \geq 2 $, we need to consider two cases due to the absolute value:

$ 2x + 3 \geq 2 $
$ 2x + 3 \leq -2 $

We will solve each inequality separately to find the range of values for $ x $.

Step 1: Set Up the Inequalities

To solve the inequality $ |2x + 3| \geq 2 $, we consider two cases:

$ 2x + 3 \geq 2 $
$ 2x + 3 \leq -2 $

Step 2: Solve the First Inequality

For the first inequality $ 2x + 3 \geq 2 $, we solve for $ x $: \[ 2x + 3 \geq 2 \implies 2x \geq -1 \implies x \geq -\frac{1}{2} \]

Step 3: Solve the Second Inequality

For the second inequality $ 2x + 3 \leq -2 $, we solve for $ x $: \[ 2x + 3 \leq -2 \implies 2x \leq -5 \implies x \leq -\frac{5}{2} \]

Step 4: Combine the Solutions

The solution to the inequality $ |2x + 3| \geq 2 $ is the union of the solutions from the two cases: \[ x \geq -\frac{1}{2} \quad \text{or} \quad x \leq -\frac{5}{2} \]

Final Answer

The solution to the inequality is: \[ \boxed{x \in \left(-\infty, -\frac{5}{2}\right] \cup \left[-\frac{1}{2}, \infty\right)} \]

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