Absolute Value Equations 47F074

1. **Problem:** Solve $|x - 9| = 72$. **Step 1:** Recall that $|A| = B$ means $A = B$ or $A = -B$ if $B \geq 0$. **Step 2:** Set up two equations: $$x - 9 = 72$$ $$x - 9 = -72$$ **Step 3:** Solve each: $$x = 72 + 9 = 81$$ $$x = -72 + 9 = -63$$ **Answer:** $x = 81$ or $x = -63$. 2. **Problem:** Solve $|3x - 6| = 18$. **Step 1:** Set up two equations: $$3x - 6 = 18$$ $$3x - 6 = -18$$ **Step 2:** Solve each: $$3x = 18 + 6 = 24 \Rightarrow x = \frac{24}{3} = 8$$ $$3x = -18 + 6 = -12 \Rightarrow x = \frac{-12}{3} = -4$$ **Answer:** $x = 8$ or $x = -4$. 3. **Problem:** Solve $|10x + 30| = 0$. **Step 1:** Absolute value equals zero only if the inside is zero: $$10x + 30 = 0$$ **Step 2:** Solve: $$10x = -30 \Rightarrow x = \frac{-30}{10} = -3$$ **Answer:** $x = -3$. 4. **Problem:** Solve $|x + 56| = -156$. **Step 1:** Absolute value cannot be negative. **Answer:** No solution. 5. **Problem:** Solve $|2x + 5| + 8 = 23$. **Step 1:** Isolate absolute value: $$|2x + 5| = 23 - 8 = 15$$ **Step 2:** Set up two equations: $$2x + 5 = 15$$ $$2x + 5 = -15$$ **Step 3:** Solve each: $$2x = 15 - 5 = 10 \Rightarrow x = \frac{10}{2} = 5$$ $$2x = -15 - 5 = -20 \Rightarrow x = \frac{-20}{2} = -10$$ **Answer:** $x = 5$ or $x = -10$. 6. **Problem:** Solve $|x^2 + 9x + 14| = 0$. **Step 1:** Absolute value zero means inside zero: $$x^2 + 9x + 14 = 0$$ **Step 2:** Factor quadratic: $$x^2 + 9x + 14 = (x + 7)(x + 2) = 0$$ **Step 3:** Solve each factor: $$x + 7 = 0 \Rightarrow x = -7$$ $$x + 2 = 0 \Rightarrow x = -2$$ **Answer:** $x = -7$ or $x = -2$. 7. **Problem:** Solve $|x^2 - 4| = -4$. **Step 1:** Absolute value cannot be negative. **Answer:** No solution. 8. **Problem:** Solve $|x - 7| + 2 = 2$. **Step 1:** Isolate absolute value: $$|x - 7| = 2 - 2 = 0$$ **Step 2:** Absolute value zero means inside zero: $$x - 7 = 0 \Rightarrow x = 7$$ **Answer:** $x = 7$.