Question

Solve the given inequalities and represent the solution graphically on the number line: \[ 2(x - 1)<x + 5, \quad 3(x + 2)>2 - x \]

Hint:

When solving compound inequalities, solve each inequality separately and then find the intersection of the solutions.

Answer 1

Step 1: Solve the first inequality.
The first inequality is: \[ 2(x - 1)<x + 5 \] Simplify the left side: \[ 2x - 2<x + 5 \] Subtract \( x \) from both sides: \[ x - 2<5 \] Add 2 to both sides: \[ x<7 \]
Step 2: Solve the second inequality.
The second inequality is: \[ 3(x + 2)>2 - x \] Simplify the left side: \[ 3x + 6>2 - x \] Add \( x \) to both sides: \[ 4x + 6>2 \] Subtract 6 from both sides: \[ 4x>-4 \] Divide by 4: \[ x>-1 \]
Step 3: Represent the solution on the number line.
The solution to the system of inequalities is: \[ -1<x<7 \] On the number line, the solution is represented by an open interval between -1 and 7.