Questions: Which of the following represents this inequality? 4x+5 ≥ 6

Which of the following represents this inequality? 4x+5 ≥ 6
Transcript text: Which of the following represents this inequality? \[ |4 x+5| \geq 6 \]
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Solution

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Solution Steps

Step 1: Understand the Inequality

The given inequality is \( |4x + 5| \geq 6 \). This means the absolute value of \( 4x + 5 \) is greater than or equal to 6.

Step 2: Break Down the Absolute Value Inequality

To solve \( |4x + 5| \geq 6 \), we need to consider two cases:

  1. \( 4x + 5 \geq 6 \)
  2. \( 4x + 5 \leq -6 \)
Step 3: Solve Each Case

  1. For \( 4x + 5 \geq 6 \): \[ 4x + 5 \geq 6 \\ 4x \geq 1 \\ x \geq \frac{1}{4} \]

  2. For \( 4x + 5 \leq -6 \): \[ 4x + 5 \leq -6 \\ 4x \leq -11 \\ x \leq -\frac{11}{4} \]

Step 4: Combine the Solutions

The combined solution is: \[ x \leq -\frac{11}{4} \quad \text{or} \quad x \geq \frac{1}{4} \]

Step 5: Identify the Correct Graph

The correct graph will show two separate intervals: one starting from \(-\frac{11}{4}\) and extending to the left, and another starting from \(\frac{1}{4}\) and extending to the right.

Final Answer

The correct graph is option (a).

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