1. **Problem:** Solve the linear inequality $$\frac{2x}{3} - \frac{1}{2} < 0$$. 2. **Formula and rules:** To solve inequalities, isolate the variable on one side. Remember, multiplying or dividing by a negative number reverses the inequality sign. 3. **Work:** $$\frac{2x}{3} - \frac{1}{2} < 0$$ Add $$\frac{1}{2}$$ to both sides: $$\frac{2x}{3} < \frac{1}{2}$$ Multiply both sides by 3: $$2x < \frac{3}{2}$$ Divide both sides by 2: $$x < \frac{3}{4}$$ 4. **Answer:** The solution is $$x < \frac{3}{4}$$. --- 1. **Problem:** Solve the inequality $$y \geq x - 1$$. 2. **Formula and rules:** This is a linear inequality in two variables. The boundary line is $$y = x - 1$$. The solution includes points on or above this line. 3. **Answer:** The solution set is all points $$ (x,y) $$ such that $$ y \geq x - 1 $$. --- 1. **Problem:** Solve the inequality $$3x_1 - 5x_2 \leq 2$$. 2. **Formula and rules:** This inequality represents a half-plane bounded by the line $$3x_1 - 5x_2 = 2$$. 3. **Answer:** The solution set is all points $$ (x_1,x_2) $$ satisfying $$3x_1 - 5x_2 \leq 2$$. --- 1. **Problem:** Solve the system: $$5x_1 + 4x_2 \leq 40$$ $$x_1 + 2x_2 \leq 14$$ 2. **Formula and rules:** Each inequality defines a half-plane. The solution is the intersection of these half-planes. 3. **Answer:** The solution set is all $$ (x_1,x_2) $$ satisfying both inequalities. --- 1. **Problem:** Solve the system: $$4x + 3y > 24$$ $$x + y \leq 8$$ 2. **Formula and rules:** The first inequality is strict, so the solution excludes the boundary line. 3. **Answer:** The solution set is all $$ (x,y) $$ such that $$4x + 3y > 24$$ and $$x + y \leq 8$$. --- 1. **Problem:** Solve the system of linear equations by elimination: $$7x - 3y = -14$$ $$-3x + y = 6$$ 2. **Formula and rules:** Multiply the second equation to align coefficients for elimination. 3. **Work:** Multiply second equation by 3: $$-9x + 3y = 18$$ Add to first equation: $$7x - 3y + (-9x + 3y) = -14 + 18$$ $$-2x = 4$$ $$x = -2$$ Substitute $$x = -2$$ into second equation: $$-3(-2) + y = 6$$ $$6 + y = 6$$ $$y = 0$$ 4. **Answer:** $$x = -2, y = 0$$. --- 1. **Problem:** Solve the system: $$4x + 5y = 3$$ $$3x + y = 5$$ 2. **Work:** Multiply second equation by 5: $$15x + 5y = 25$$ Subtract first equation: $$15x + 5y - (4x + 5y) = 25 - 3$$ $$11x = 22$$ $$x = 2$$ Substitute $$x=2$$ into second equation: $$3(2) + y = 5$$ $$6 + y = 5$$ $$y = -1$$ 3. **Answer:** $$x=2, y=-1$$. --- 1. **Problem:** Solve the system: $$3x + 12y = 72$$ $$x + 4y = 24$$ 2. **Work:** Multiply second equation by 3: $$3x + 12y = 72$$ Subtract first equation: $$3x + 12y - (3x + 12y) = 72 - 72$$ $$0=0$$ (dependent system) 3. **Answer:** Infinite solutions along the line $$x + 4y = 24$$. --- 1. **Problem:** Solve the system: $$4x + 2y = 8$$ $$2x + y = -12$$ 2. **Work:** Multiply second equation by 2: $$4x + 2y = -24$$ Subtract first equation: $$4x + 2y - (4x + 2y) = -24 - 8$$ $$0 = -32$$ (contradiction) 3. **Answer:** No solution; the system is inconsistent.