교과서 질문
Decide whether each statement is true or false. The solution set of 2x+5=x -3 is {-8}.
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Ch. 1 - Equations and Inequalities
Lial13th EditionCollege AlgebraISBN: 9780136881063
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2장, 문제 7
Solve each problem. If x represents the number of pennies in a jar in an applied problem, which of the following equations cannot be a correct equation for finding x? (Hint:Solve the equations and consider the solutions.)
A. 5x+3 =11
B.12x+6 =-4
C.100x =50(x+3)
D. 6(x+4) =x+24
검증된 단계별 안내1
Step 1: Understand the context of the problem. Since \( x \) represents the number of pennies in a jar, \( x \) must be a non-negative integer (\( x \geq 0 \)) because you cannot have a negative number of pennies.
Step 2: Solve each equation for \( x \) to find its solution.
For equation A: \( 5x + 3 = 11 \), subtract 3 from both sides to get \( 5x = 8 \), then divide both sides by 5 to find \( x = \frac{8}{5} \).
For equation B: \( 12x + 6 = -4 \), subtract 6 from both sides to get \( 12x = -10 \), then divide both sides by 12 to find \( x = -\frac{10}{12} = -\frac{5}{6} \).
For equation C: \( 100x = 50(x + 3) \), expand the right side to get \( 100x = 50x + 150 \), subtract \( 50x \) from both sides to get \( 50x = 150 \), then divide both sides by 50 to find \( x = 3 \).
For equation D: \( 6(x + 4) = x + 24 \), expand the left side to get \( 6x + 24 = x + 24 \), subtract \( x \) and 24 from both sides to get \( 5x = 0 \), then divide both sides by 5 to find \( x = 0 \).
Step 3: Analyze the solutions in the context of the problem. Since \( x \) must be non-negative, any negative solution is not valid for the number of pennies.
Step 4: Identify which equation(s) yield invalid solutions. Equation B gives \( x = -\frac{5}{6} \), which is negative and therefore cannot represent the number of pennies.
Step 5: Conclude that equation B cannot be a correct equation for finding \( x \) in this applied problem.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Solving Linear Equations
Linear equations are algebraic expressions where the variable is to the first power. Solving them involves isolating the variable on one side to find its value. Understanding how to manipulate equations using addition, subtraction, multiplication, and division is essential to find solutions.
추천 영상:
04:02
Solving Linear Equations with Fractions
Interpreting Solutions in Context
When variables represent real-world quantities, like the number of pennies, solutions must make sense in that context. For example, negative or non-integer solutions may be invalid if the quantity cannot be negative or fractional. Evaluating solutions against the problem's scenario ensures meaningful answers.
추천 영상:
2:57Probability of Non-Mutually Exclusive Events Example
Checking for Extraneous or Invalid Solutions
Some equations may yield solutions that do not fit the problem's conditions or are mathematically impossible. Identifying such solutions requires substituting back or analyzing the equation's structure. This helps determine which equations cannot correctly represent the situation.
추천 영상:
05:21Restrictions on Rational Equations
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