1. Solve the inequality $\frac{2x}{3} - \frac{1}{2} < 0$. Step 1: Add $\frac{1}{2}$ to both sides: $$\frac{2x}{3} < \frac{1}{2}$$ Step 2: Multiply both sides by 3 to clear the denominator: $$\cancel{3} \times \frac{2x}{\cancel{3}} < \frac{1}{2} \times 3$$ $$2x < \frac{3}{2}$$ Step 3: Divide both sides by 2: $$\frac{\cancel{2}x}{\cancel{2}} < \frac{3}{2} \times \frac{1}{2}$$ $$x < \frac{3}{4}$$ 2. Solve the inequality $y \geq x - 1$. This is already in slope-intercept form. The solution set is all points on or above the line $y = x - 1$. 3. Solve $3x_1 - 5x_2 \leq 2$. This inequality describes a half-plane including the line $3x_1 - 5x_2 = 2$ and all points below it. 4. Solve the system: $$5x_1 + 4x_2 \leq 40$$ $$x_1 + 2x_2 \leq 14$$ These inequalities describe the intersection of two half-planes. 5. Solve the system: $$4x + 3y > 24$$ $$x + y \leq 8$$ The solution set is the intersection of the half-plane above the line $4x + 3y = 24$ and below or on the line $x + y = 8$. II. Solve systems of linear equations by elimination. 1. Solve: $$7x - 3y = -14$$ $$-3x + y = 6$$ 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: $$-3(-2) + y = 6$$ $$6 + y = 6$$ $$y = 0$$ Solution: $(-2, 0)$ 2. Solve: $$4x + 5y = 3$$ $$3x + y = 5$$ 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: $$3(2) + y = 5$$ $$6 + y = 5$$ $$y = -1$$ Solution: $(2, -1)$ 3. Solve: $$3x + 12y = 72$$ $$x + 4y = 24$$ Multiply second equation by 3: $$3x + 12y = 72$$ Subtract first equation: $$(3x + 12y) - (3x + 12y) = 72 - 72$$ $$0 = 0$$ This means the two equations are dependent; infinite solutions along the line $x + 4y = 24$. 4. Solve: $$4x + 2y = 8$$ $$2x + y = -12$$ Multiply second equation by 2: $$4x + 2y = -24$$ Subtract first equation: $$(4x + 2y) - (4x + 2y) = -24 - 8$$ $$0 = -32$$ Contradiction, no solution. Desmos functions for graphing inequalities and equations: - $y = x - 1$ - $3x_1 - 5x_2 = 2$ - $5x_1 + 4x_2 = 40$ - $x_1 + 2x_2 = 14$ - $4x + 3y = 24$ - $x + y = 8$ - $7x - 3y = -14$ - $-3x + y = 6$ - $4x + 5y = 3$ - $3x + y = 5$ - $3x + 12y = 72$ - $x + 4y = 24$ - $4x + 2y = 8$ - $2x + y = -12$