Solve for x and sum of possible integers. **Topic: Solving Quadratic Inequalities**### Inequality:\[ 1 < (x-2)^2 < 25 \]### Explanation:The inequality \(1 < (x-2)^2 < 25\) is a compound inequality involving a quadratic expression. Here, we need to find the values of \(x\) that satisfy both \(1 < (x-2)^2\) and \((x-2)^2 < 25\) simultaneously.#### Step-by-step Solution:**Step 1: Solve \( (x-2)^2 < 25 \)**1. Take the square root on both sides: \[ -5 < x-2 < 5 \]2. Add 2 to each part of the inequality:\[ -5 + 2 < x < 5 + 2 \]\[ -3 < x < 7 \]**Step 2: Solve \( 1 < (x-2)^2 \)**1. Take the square root on both sides, noting both positive and negative roots:\[ 1 < x-2 \quad \text{or} \quad x-2 < -1 \]2. Solve each part separately: - For \(1 < x-2\): \[ x > 3 \] - For \(x-2 < -1\): \[ x < 1 \]**Step 3: Combine the solutions**- From \(1 < (x-2)^2 < 25\), we need the intersection of \( -3 < x < 7 \) and \( x > 3 \) or \( x < 1 \).So, the combined solution is:\[ -3 < x < 1 \quad \text{or} \quad 3 < x < 7 \]Thus, the solution set for the inequality \(1 < (x-2)^2 < 25\) is:\[ -3 < x < 1 \quad \text{or} \quad 3 < x < 7 \]### Graphical Representation:To visualize this, plot the solutions on a number line:1. Draw a number line.2. Mark and shade the regions between \(-3\) and \(1\) (but not including \(-3\) and \(1\)) and between \(3\) and \(7\) (

Name: Solve for x and sum of possible integers. **Topic: Solving Quadratic I
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Solve for x and sum of possible integers. **Topic: Solving Quadratic Inequalities**### Inequality:\[ 1 < (x-2)^2 < 25 \]### Explanation:The inequality \(1 < (x-2)^2 < 25\) is a compound inequality involving a quadratic expression. Here, we need to fi