Solve the equations and graph your solution: 11. $$ |2 g-5|-3 \geq 7 $$

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UpStudy AI · Accepted Answer

To solve the inequality $$ |2g - 5| - 3 \geq 7 $$ and graph the solution, follow these steps:

### Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression:

$$
|2g - 5| - 3 \geq 7
$$

Add **3** to both sides of the inequality:

$$
|2g - 5| \geq 10
$$

### Step 2: Understand the Absolute Value Inequality

The inequality $$ |A| \geq C $$ (where $$ C \geq 0 $$) means that:

$$
A \leq -C \quad \text{or} \quad A \geq C
$$

Applying this to our equation:

$$
2g - 5 \leq -10 \quad \text{or} \quad 2g - 5 \geq 10
$$

### Step 3: Solve Each Inequality Separately

**First Inequality:**

$$
2g - 5 \leq -10
$$

Add **5** to both sides:

$$
2g \leq -5
$$

Divide by **2**:

$$
g \leq -2.5
$$

**Second Inequality:**

$$
2g - 5 \geq 10
$$

Add **5** to both sides:

$$
2g \geq 15
$$

Divide by **2**:

$$
g \geq 7.5
$$

### Step 4: Combine the Solutions

The solutions from both inequalities are:

$$
g \leq -2.5 \quad \text{or} \quad g \geq 7.5
$$

### Step 5: Graph the Solution

To graph the solution on a number line:

1. **Draw a horizontal line** representing all real numbers.
2. **Mark the critical points** at $$ g = -2.5 $$ and $$ g = 7.5 $$.
3. **Use closed circles** at $$ -2.5 $$ and $$ 7.5 $$ to indicate that these points are included in the solution (since the inequality is $$ \geq $$).
4. **Shade the regions** to the **left** of $$ -2.5 $$ and to the **right** of $$ 7.5 $$.

#### Graph Representation:

```
<----|=========|==================|=========|---->
    -∞        -2.5               7.5       ∞
```

- The shaded areas are:
  - All numbers **less than or equal to -2.5**.
  - All numbers **greater than or equal to 7.5**.

### Final Answer

The solutions are all real numbers g with g ≤ –2.5 or g ≥ 7.5.

**Graphically:**

• Draw a number line.
• Shade all numbers to the left of –2.5 and to the right of 7.5.
• Include closed circles at –2.5 and 7.5.

Thus, the solution is:
$$ g \leq -2.5 \quad \text{or} \quad g \geq 7.5 $$
The solutions are all real numbers $$ g $$ such that $$ g \leq -2.5 $$ or $$ g \geq 7.5 $$.

Solve the equations and graph your solution: 11. \( |2 g-5|-3 \geq 7 \)

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