Problem 53 - College Algebra Essentials

Question

Solve each compound inequality. $$-3 \leq x-2<1$$

Step-by-Step Solution

Answer

The solution for the compound inequality \(-3 \leq x-2<1\) is \(-1 \leq x < 3\).

Step 1: Divide the compound inequality

The compound inequality is divided into two simple inequalities. These are: \(-3 \leq x-2\) and \(x-2<1\).

Step 2: Solve the first inequality

The first inequality is \(-3 \leq x-2\). To solve for \(x\), isolate \(x\) by adding 2 to both sides. That gives \(-3+2 \leq x\), hence \(x \geq -1\)

Step 3: Solve the second inequality

The second inequality is \(x-2<1\). To solve for \(x\), again add 2 to both sides. That gives \(x-2+2<1+2\), hence \(x<3\).

Step 4: Combine the solutions

The solutions from the two inequalities can be combined to form the solution to the original compound inequality. Since this compound inequality has both values \(x\) can be all the real numbers between -1 and 3 including -1 but less than 3. Therefore, the solution is \(-1 \leq x < 3\).