DQ. 1 Can a linear equation and a linear inequality be solved in the same way? Explain why? What makes them different?
No, a linear equation and a linear inequality cannot be solved in the same way because inequality has more than one answer whereas linear equation has at most one answer. If you multiply or divide both sides by a negative number, then the sign of the inequality is reversed whereas multiplication or division by negative number in linear equation has no effect on the sign. If you take reciprocals of both sides then also the sign of the inequality is reversed.
Example: consider the linear equation, -1/x = ½.
Reciprocal: -x = 2,
Multiplication by -1: x= -2
Now, consider the linear inequality, -1/x < ½
Reciprocal: -x > 2
Multiplication by -1: x < -2.
DQ 2. What are the four steps for solving an equation? Should any other factors be accounted for when solving an equation? Should any factors be accounted for when explaining how to solve an equation? Explain your answer.
The following are the four steps for solving an equation:
1) Using distribution property, add or subtract any like terms on both sides.
2) Using addition property of the equality, brings variable terms on one side and constant terms on the other sides.
3) Using the multiplication property of the equality, multiply both sides by the inverse of the coefficient of the variable.
4) Put the value of variable in original equation to check the answer.
Example: Consider the equation 3x + 7 = 21 – 2x + 3x
Step 1: 3x + 7 = 21 – 2x + 3x è 3x + 7 = 21 + x
Step 2: 3x + 7 = 21 + x è 3x + 7 – x – 7 = 21 + x – x – 7 è 2x = 14
Step 3: 2x = 14 è (1/2)(2x) = (1/2)(14) è x = 7
Step 4: put x=7 in 3x+7 = 21 – 2x + 3x è 3*7 + 7 = 21 – 2*7 + 3*7
è 21 + 7 = 21 – 14 + 21 è 28 = 28
These are the only factors one should consider when solving an equation.
While explaining how to solve an equation one should considers the order of operations to be executed. Here are the rank of operations: parentheses, Exponents, multiplication, division, addition and the subtraction.