1. **Problem 1: Solve the linear equation** Solve for $x$ in the equation $$3x - 7 = 2x + 5$$. 2. **Problem 2: Solve the linear inequality** Solve for $x$ in the inequality $$4x + 1 > 3x - 2$$. 3. **Problem 3: Draw the graph of the linear equation** Graph the equation $$y = 2x - 3$$. 4. **Problem 4: Draw the graph of the linear inequality** Graph the inequality $$y \\leq -x + 4$$. 5. **Problem 5: Real-life application** A taxi company charges a base fare of 5 plus 2 per kilometer. Write a linear equation for the total fare $y$ based on kilometers $x$. Then, find the fare for 10 kilometers and graph the equation. --- **Formulas and rules:** - To solve linear equations, isolate $x$ by performing inverse operations. - For inequalities, remember to reverse the inequality sign when multiplying or dividing by a negative number. - The graph of $y = mx + b$ is a straight line with slope $m$ and y-intercept $b$. - For inequalities, shade the region above or below the line depending on the inequality sign. --- **Step-by-step solutions:** **1. Solve $3x - 7 = 2x + 5$** - Subtract $2x$ from both sides: $3x - 2x - 7 = 5$ - Simplify: $x - 7 = 5$ - Add 7 to both sides: $x = 12$ **2. Solve $4x + 1 > 3x - 2$** - Subtract $3x$ from both sides: $4x - 3x + 1 > -2$ - Simplify: $x + 1 > -2$ - Subtract 1 from both sides: $x > -3$ **3. Graph $y = 2x - 3$** - Y-intercept is $-3$ (point $(0,-3)$) - Slope is $2$ (rise over run: up 2, right 1) - Plot points $(0,-3)$ and $(1,-1)$ and draw a straight line through them. **4. Graph $y \\leq -x + 4$** - Y-intercept is $4$ (point $(0,4)$) - Slope is $-1$ (down 1, right 1) - Plot points $(0,4)$ and $(1,3)$ - Draw a solid line (because of $\leq$) and shade below the line. **5. Real-life problem** - Equation: $y = 2x + 5$ - For $x=10$: $y = 2(10) + 5 = 20 + 5 = 25$ - Graph points $(0,5)$ and $(10,25)$ and draw the line. These problems cover solving equations and inequalities, graphing, and applying concepts to real life.