Solve the following Inequalities and the represent soldston on a namber line 1) \( x+18 \leq 9-2 x \) 3) \( 5(x-1)>1(x-1) \) 3) \( \frac{x}{3}-\frac{x}{3}>1 \) 4) \( -8-<3 x-2-4 \)

1) Solve \( x+18 \leq 9-2x \). - Add \(2x\) to both sides: \[ x + 18 + 2x \leq 9 \] - Combine like terms: \[ 3x + 18 \leq 9 \] - Subtract 18 from both sides: \[ 3x \leq 9 - 18 \quad \Longrightarrow \quad 3x \leq -9 \] - Divide by 3: \[ x \leq -3 \] Number line representation: a ray extending to the left from \(-3\) with a closed circle at \(-3\). --- 2) Solve \( 5(x-1) > 1(x-1) \). - Subtract \(1(x-1)\) from both sides: \[ 5(x-1) - 1(x-1) > 0 \] - Factor the common term: \[ (5-1)(x-1) > 0 \quad \Longrightarrow \quad 4(x-1) > 0 \] - Divide by 4 (positive, so the inequality remains in the same direction): \[ x-1 > 0 \quad \Longrightarrow \quad x > 1 \] Number line representation: a ray extending to the right from \(1\) with an open circle at \(1\). --- 3) Solve \( \frac{x}{3}-\frac{x}{3} > 1 \). - Simplify the left-hand side: \[ \frac{x}{3} - \frac{x}{3} = 0 \] - The inequality becomes: \[ 0 > 1 \] Since \(0 > 1\) is false for all real numbers, there is no solution. Number line representation: an empty set (no shading). --- 4) Solve \( -8 < 3x-2-4 \). - Simplify the right-hand side: \[ 3x -2 -4 = 3x - 6 \] - The inequality becomes: \[ -8 < 3x - 6 \] - Add 6 to both sides: \[ -8 + 6 < 3x \quad \Longrightarrow \quad -2 < 3x \] - Divide by 3: \[ -\frac{2}{3} < x \quad \Longrightarrow \quad x > -\frac{2}{3} \] Number line representation: a ray extending to the right from \(-\frac{2}{3}\) with an open circle at \(-\frac{2}{3}\).