Solve the inequalities and represent the solution graphically on number line:
The solution set is x ∈ (-1, 7).
Explanation
This is a system of linear inequalities. We solve each inequality separately and then find the intersection of both solutions. Following the method shown in Example 11 of the textbook, we solve both inequalities and combine the results.
Solution Steps
Step 1: Solve the first inequality 2(x - 1) < x + 5 Opening brackets: 2x - 2 < x + 5 Transposing terms: 2x - x < 5 + 2 Simplifying: x < 7
Step 2: Solve the second inequality 3(x + 2) > 2 - x Opening brackets: 3x + 6 > 2 - x Transposing terms: 3x + x > 2 - 6 Simplifying: 4x > -4 Dividing by 4: x > -1
Step 3: Find the common solution From first inequality: x < 7 From second inequality: x > -1 The values of x satisfying both inequalities are: -1 < x < 7 Therefore, the solution set is x ∈ (-1, 7).
Step 4: Graphical representation on number line Draw a number line. Mark points -1 and 7. Draw an open circle at -1 and 7 (since endpoints are not included). Shade the region between -1 and 7 to represent all real numbers greater than -1 and less than 7.