$ To isolate $x$, add 4 to both sides: $$x - 4 + 4 = 9 + 4$$ $$x = 13$$ **Check:** $13 - 4 = 9$ ✓ ## Multiplication/Division Equations ### Example 3: $3x = 15$ To isolate $x$, divide both sides by 3: $$\frac{3x}{3} = \frac{15}{3}$$ $$x = 5$$ **Check:** $3(5) = 15$ ✓ ### Example 4: $\frac{x}{4} = 6$ To isolate $x$, multiply both sides by 4: $$4 \cdot \frac{x}{4} = 4 \cdot 6$$ $$x = 24$$ **Check:** $\frac{24}{4} = 6$ ✓ ## Practice Strategy 1. Identify what operation is applied to $x$ 2. Apply the inverse operation to both sides 3. Simplify to find $x$ 4. Check your answer!27:T467,# Two-Step Equations Two-step equations require two inverse operations to solve. ## General Strategy The standard form is: $ax + b = c$ **Steps:** 1. Undo addition/subtraction first 2. Then undo multiplication/division This is the reverse of order of operations! ## Examples ### Example 1: $2x + 5 = 13$ **Step 1:** Subtract 5 from both sides $$2x + 5 - 5 = 13 - 5$$ $$2x = 8$$ **Step 2:** Divide both sides by 2 $$\frac{2x}{2} = \frac{8}{2}$$ $$x = 4$$ **Check:** $2(4) + 5 = 8 + 5 = 13$ ✓ ### Example 2: $\frac{x}{3} - 4 = 2$ **Step 1:** Add 4 to both sides $$\frac{x}{3} - 4 + 4 = 2 + 4$$ $$\frac{x}{3} = 6$$ **Step 2:** Multiply both sides by 3 $$3 \cdot \frac{x}{3} = 3 \cdot 6$$ $$x = 18$$ **Check:** $\frac{18}{3} - 4 = 6 - 4 = 2$ ✓ ### Example 3: $-3x + 7 = -2$ **Step 1:** Subtract 7 from both sides $$-3x + 7 - 7 = -2 - 7$$ $$-3x = -9$$ **Step 2:** Divide both sides by -3 $$\frac{-3x}{-3} = \frac{-9}{-3}$$ $$x = 3$$ **Check:** $-3(3) + 7 = -9 + 7 = -2$ ✓ ## Summary 1. Deal with constants first (add/subtract) 2. Deal with coefficients second (multiply/divide) 3. Always check your answer!28:T502,# Introduction to Inequalities While equations use $=$, inequalities compare values that are **not** equal. ## Inequality Symbols | Symbol | Meaning | Example | |--------|---------|---------| | $<$ | Less than | $3 < 5$ | | $>$ | Greater than | $7 > 4$ | | $\leq$ | Less than or equal to | $x \leq 3$ | | $\geq$ | Greater than or equal to | $y \geq -2$ | ## Reading Inequalities **Tip:** The "hungry alligator" always eats the bigger number! - $x < 5$ reads as "x is less than 5" - $x > 5$ reads as "x is greater than 5" - $x \leq 5$ reads as "x is less than or equal to 5" (x can be 5!) - $x \geq 5$ reads as "x is greater than or equal to 5" (x can be 5!) ## Solutions to Inequalities Unlike equations with one solution, inequalities have **many solutions**. For $x > 3$: - $x = 4$ ✓ - $x = 5$ ✓ - $x = 100$ ✓ - $x = 3.001$ ✓ - $x = 3$ ✗ (not greater than 3) - $x = 2$ ✗ ## Graphing on a Number Line We show solutions on a number line: - **Open circle** (○): Number is NOT included ($<$ or $>$) - **Closed circle** (●): Number IS included ($\leq$ or $\geq$) - **Arrow**: Shows direction of all solutions ### Example: $x > 2$ Use an open circle at 2, arrow pointing right → ### Example: $x \leq -1$ Use a closed circle at -1, arrow pointing left ←29:T4c2,# Solving Inequalities Solving inequalities is similar to solving equations, with one important exception! ## Same Rules Apply You can: - Add/subtract the same number from both sides - Multiply/divide both sides by the same **positive** number ### Example 1: $x + 4 < 9$ Subtract 4 from both sides: $$x + 4 - 4 < 9 - 4$$ $$x < 5$$ ### Example 2: $3x \geq 15$ Divide both sides by 3: $$\frac{3x}{3} \geq \frac{15}{3}$$ $$x \geq 5$$ ## ⚠️ The Critical Rule **When you multiply or divide by a NEGATIVE number, FLIP the inequality sign!** Why? Consider: $2 < 4$ Multiply both sides by $-1$: $-2$ and $-4$ Is $-2 < -4$? No! $-2 > -4$ ### Example 3: $-2x > 6$ Divide by $-2$ AND flip the sign: $$\frac{-2x}{-2} < \frac{6}{-2}$$ $$x < -3$$ ### Example 4: $-\frac{x}{3} \leq 4$ Multiply by $-3$ AND flip the sign: $$(-3) \cdot (-\frac{x}{3}) \geq (-3) \cdot 4$$ $$x \geq -12$$ ## Two-Step Inequalities ### Example 5: $2x - 5 > 3$ **Step 1:** Add 5 to both sides $$2x - 5 + 5 > 3 + 5$$ $$2x > 8$$ **Step 2:** Divide by 2 (positive, so no flip) $$x > 4$$

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