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

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: \[ g \leq -2.5 \quad \text{or} \quad g \geq 7.5 \]