Unit 2: Equations & Inequalities Date: Bell: Homework 5: Absolute Value Equations ** This is a 2-page document! ** Directions: Solve each equation. SHOW ALL STEPS! 1. x = 3 2. m = – 10 3.-6k = 30 4. c- 10 = 3 6. -48-2 = 10 5. 2k + 3 =7 8. 26- 4c =-2 5x +5 =13 Gina Wilson (All Things Algebra, LLC), 2012-2016
The correct answer and explanation is:
Here’s the solution to the equations and a detailed explanation of the process.
Solutions
- (x = 3)
This equation is already solved. The solution is (x = 3). - (m = -10)
This equation is also already solved. The solution is (m = -10). - (-6k = 30)
To solve for (k):
[
k = \frac{30}{-6} = -5
]
Solution: (k = -5) - (c – 10 = 3)
Add 10 to both sides:
[
c = 3 + 10 = 13
]
Solution: (c = 13) - (2k + 3 = 7)
Subtract 3 from both sides:
[
2k = 4
]
Divide by 2:
[
k = \frac{4}{2} = 2
]
Solution: (k = 2) - (-48 – 2 = 10)
This appears to be incomplete or miswritten. A valid equation is required to solve. - (5x + 5 = 13)
Subtract 5 from both sides:
[
5x = 8
]
Divide by 5:
[
x = \frac{8}{5}
]
Solution: (x = \frac{8}{5}) - (26 – 4c = -2)
Subtract 26 from both sides:
[
-4c = -2 – 26
]
[
-4c = -28
]
Divide by -4:
[
c = \frac{-28}{-4} = 7
]
Solution: (c = 7)
Explanation
Solving equations involves isolating the variable on one side of the equation. For simple equations like (x = 3) and (m = -10), the variables are already isolated, so no further work is needed.
In equations like (-6k = 30), the variable is multiplied by a coefficient. To solve, divide both sides of the equation by the coefficient (-6 in this case). For (c – 10 = 3), the variable is combined with a constant. Adding 10 to both sides isolates (c). Similarly, (2k + 3 = 7) requires two steps: subtract 3 from both sides and then divide by 2.
Equations like (5x + 5 = 13) and (26 – 4c = -2) follow similar principles but involve slightly more complex arithmetic. For (5x + 5 = 13), subtracting 5 simplifies the equation, and dividing by 5 isolates (x). In (26 – 4c = -2), subtracting 26 first moves the constant to the other side, simplifying the equation to (-4c = -28). Dividing by -4 gives the solution.
Each step in solving equations maintains the balance of the equation by performing the same operation on both sides. This ensures the solution is accurate. Always verify the solution by substituting it back into the original equation. This process helps reinforce the solution’s correctness and builds confidence in problem-solving.