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Section 6.1
Solving Inequalities
Pass Skills 2.3a
Algebra I
Obj: state and use symbols of inequality.
Obj: solve inequalities that involve addition and subtraction.
Vocab
Addition property of inequality  let a, b, and c be real #’s. If a <b, then a + c < b + c. Adding equal
amounts to each side of an equation results in an equivalent equation.
Inequality  a statement that 2 expressions are not equal. Contain the following signs: <,>,,, 
Subtraction property of inequality  let a, b, and c be real #’s. If a < b, then a – c < b – c. If equal
amounts are subtracted from the expresseions on each side of an inequality the resulting inequality is
true.
Statements of Inequality
a is less than b.
a<b
a is greater than b.
a>b
a is less than or equal to b.
ab
a is greater than or equal to b.
ab
a is greater than b and less than c.
b<a<c
a is greater than or equal to b and
less than or equal to c.
bac
a is not equal to b.
ab
Solve inequalities, and graph the solution on a number line.
x + 12  16
x + 12 -12  16 -12
x4
Section 6.2
Multistep Inequalities
Algebra I
Obj: state and apply the multiplication and division properties of inequality.
Obj: solve multistep inequalities in one variable.
Vocab
Division property of inequality  let a, b, and c be nonzero real #’s. For c > 0, if a > b, then a/c > b/c, and
if a < b, then a/c > b/c. For c < 0, if a < b, then a/c > b/c and if a > b then a/c < b/c
Multiplication property of inequality  let a, b, and c be nonzero real #’s. For c > 0, if a > b, then ac > bc,
and if a < b, then ac < bc. For c < 0, if a < b, then ac> bc, and if a > b, then ac < bc.
Summary of Multiplication and Division Properties of Inequalities
ACTION
RESULT
Multiply or divide by a positive #
inequality sign stays the same
Multiply or divide by a negative #
inequality sign must be reversed
Solve 18 – 2y > 2.
18 – 2y > 2
18 – 18 – 2y > 2 – 18
–2y > –16
-2y/-2 < -16/-2 (Reverse inequality sign. )
Y<8
Section 6.3
Compound Inequalities
Algebra I
Obj: graph the solution sets of compound inequalities.
Obj: solve compound inequalities.
Vocab
Compound inequality  2 inequalities that are combined into one statement by the word and or or.
Conjunction  a compound inequality whose solution region is an intersection.
Disjunction  a compound inequality whose solution region is outside an intersection.
Intersection (of graphs) the solution to a system of linear equalities or inequalities, consisting of the
solutions common to each.
Union  the union of 2 sets consists of all elements from both sets. The logical relationship OR
represents the union of sets.
Solve and graph 6 < 9y  3  24.
6 < 9y  3 and 9y – 3  24
1< y and y  3
conjunction: graph lies within the endpoints
Section 6.4
Absolute Value Functions
Algebra I
Pass Skills 2.2b,2.2e
Obj: explore features of the absolute value function.
Obj: explore basic transformations of the absolute value function.
Vocab
Absolute value  the absolute value of a real # x is the distance from x to 0 on a # line; the symbol |x|
means the absolute value of x.
Absolute value function  a function written in the form y = |x| or y = ABS(x)
Line of reflection  the line across which a graph is reflected.
Parent function  the most basic function of a family of functions, or the original function before a
transformation is applied.
Transformation  a variation such as a stretch, reflection, or translation of a parent function.
Translation  a transformation that shifts the graph of a function horizontally or vertically.
Reflection  a transformation that creates a mirror image of a given function.
Rules for absolute value.
|a| = a, for a  0
|-a| = a, for a  0
|a| = –a, for a < 0
Find the absolute value of an expression.
Find |9 – 2|.
|9 – 2| = |7|
=7
Find |2 – 9|.
|2 – 9| = |– 7|
=7
Find the domain and range of y = 4|x|.
domain: all real numbers
range: all non-negative numbers
Section 6.5 Absolute Value Equations and Inequalities
Algebra I
Pass skills 2.2e
Obj: solve absolute value equations.
Obj: solve absolute value inequalities and express the solution as a range of values on a number line.
Vocab
Absolute error the absolute value of the difference between the actual measure and the specified
measure.
Absolute value equation  an equation that includes an absolute value; it will have 2 solutions.
Absolute value inequality  an inequality that includes an absolute value.
Error  the difference between the actual measure and the specified measure.
Solve absolute-value equations.
Solve |x + 3|= 7.
Case 1:
Case 2:
x + 3 is positive
x + 3 is negative
x+3=7
x + 3 = –7
x =4
x = –10
solutions: x = –10 and x = 4
Solve absolute-value inequalities.
Solve |x – 4|  5.
Case 1:
Case 2:
x – 4 is positive
x – 4 is negative
x–45
x – 4  –5
x9
x  –1
solution: –1  x  9