Expanding the walls of our classroom. This is an interactive learning ecology for students and parents in our Algebra 2 class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
Addition Property of Inequality: For all real numbers A,B, and C, the inequalities A<B and A+C<B+C are equivalent. In words, adding the same number to each side of an inequality does not change the solution set.
In words, each side of an inequality may be multiplied (or divided) by a positive number without changing the direction of the inequality symbol. Multiplying (or dividing) by a negative number requires that we reverse the inequality symbol.
Solving a Linear Inequality: Step 1. Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed. Step 2. Isolate the variable terms on one side. use the addition property of inequality to get all terms with variables on one side of the inequality and all numbers on the other side. Step 3. Isolate the variable. use the multiplication property of inequality to change the inequality to the form xk
Intersection of Sets: For any two sets A and B, the intersection of A and B, symbolized A ∩ B, is defined as follows: A ∩ b = {x⎮x is an element of B}.
Solving a Compound Inequality with and: Step 1. Solve each inequality in the compound inequality individually. Step 2. Since the inequalities are joined with and, the solution set of the compound inequalities will include all numbers that satisfy both inequalities in Step 1. (the intersection of the solution sets.)
Union of Sets: For any two sets A and B, the union of A and B, symbolize A ∪ B, is defined as follows: A ∪ B = {x⎮x is an element of A or x is an element of B}.
Solving a Compound inequality with or: Step 1. Solve each inequality in the compound inequality individually. Step 2. Since the inequalities are joined with or, the solution set includes all numbers that satisfy either one of the two inequalities in Step 1 (the union of the solution sets).
Solving Absolute Value Equations and Inequalities. Let k be a positive real number, and p and q be real numbers. 1. To solve ⎮ax+b⎮=k, solve the compound equation ax+b=k or ax+b= -k The solution set is usually the form {p,q}, which includes two numbers
2. To solve⎮ax +b⎮>k, solve the compound inequality ax+b>k or ax+b<-k The solution set is of the form (-∞, p) ∪ (q,∞), which consists of two separate intervals.
3. To solve ⎮ax+b⎮<k, solve the three part inequality -k<ax+b<k The solution set is of the form (p.q), a single interval.
Special Cases for Absolute Value 1. The absolute value of an expression can never be negative. 2. The absolute value of an expression equals 0 only when the expression is equal to 0.
KEY TERMS FOR CHAPTER 3! interval interval notation inequality linear inequality in one variable equivalent inequalities intersection compound inequality union absolute value equation absolute value inequality
3.1: Linear Equations in One Variable
ReplyDeleteLinear Inequality: A linear inequality in one variable can be written in the form Ax+B<C where A,B, and C are real numbers, with A not equal to 0.
Addition Property of Inequality: For all real numbers A,B, and C, the inequalities A<B and A+C<B+C are equivalent. In words, adding the same number to each side of an inequality does not change the solution set.
ReplyDeleteMultiplication Property of Inequality
ReplyDeleteIn words, each side of an inequality may be multiplied (or divided) by a positive number without changing the direction of the inequality symbol. Multiplying (or dividing) by a negative number requires that we reverse the inequality symbol.
Solving a Linear Inequality:
ReplyDeleteStep 1. Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed.
Step 2. Isolate the variable terms on one side. use the addition property of inequality to get all terms with variables on one side of the inequality and all numbers on the other side.
Step 3. Isolate the variable. use the multiplication property of inequality to change the inequality to the form xk
3.2: Set Operations and Compound Inequalities
ReplyDeleteIntersection of Sets:
For any two sets A and B, the intersection of A and B, symbolized A ∩ B, is defined as follows: A ∩ b = {x⎮x is an element of B}.
Solving a Compound Inequality with and:
ReplyDeleteStep 1. Solve each inequality in the compound inequality individually.
Step 2. Since the inequalities are joined with and, the solution set of the compound inequalities will include all numbers that satisfy both inequalities in Step 1. (the intersection of the solution sets.)
Union of Sets:
ReplyDeleteFor any two sets A and B, the union of A and B, symbolize A ∪ B, is defined as follows: A ∪ B = {x⎮x is an element of A or x is an element of B}.
Solving a Compound inequality with or:
ReplyDeleteStep 1. Solve each inequality in the compound inequality individually.
Step 2. Since the inequalities are joined with or, the solution set includes all numbers that satisfy either one of the two inequalities in Step 1 (the union of the solution sets).
3.3: Absolute Value Equations and Inequalities
ReplyDeleteSolving Absolute Value Equations and Inequalities.
Let k be a positive real number, and p and q be real numbers.
1. To solve ⎮ax+b⎮=k, solve the compound equation
ax+b=k or ax+b= -k
The solution set is usually the form {p,q}, which includes two numbers
2. To solve⎮ax +b⎮>k, solve the compound inequality
ax+b>k or ax+b<-k
The solution set is of the form (-∞, p) ∪ (q,∞), which consists of two separate intervals.
3. To solve ⎮ax+b⎮<k, solve the three part inequality
-k<ax+b<k
The solution set is of the form (p.q), a single interval.
Solving ⎮ax+b⎮=⎮cx+d⎮
ReplyDeleteTo solve this absolute value, solve the compound equation
ax+b=cx+d or ax+b=-(cx+d)
Special Cases for Absolute Value
ReplyDelete1. The absolute value of an expression can never be negative.
2. The absolute value of an expression equals 0 only when the expression is equal to 0.
KEY TERMS FOR CHAPTER 3!
ReplyDeleteinterval
interval notation
inequality
linear inequality in one variable
equivalent inequalities
intersection
compound inequality
union
absolute value equation
absolute value inequality