Transcript 1.7 Introduction to Solving Inequalities

1.7 Introduction to Solving
Inequalities
Objectives: Write, solve, and graph linear
inequalities in one variable. Solve and graph
compound linear inequalities in one variable.
Standards: 2.8.11.D Formulate inequalities
to model routine and non-routine problems.
An inequality is a
mathematical statement
involving <, >, >, <, or .
Properties of Inequalities
For all real numbers a, b, and c, where a < b:
Addition Property
a + c < b + c.
Subtraction Property
a – c < b – c.
Multiplication Property
If c > 0, then ac < bc.
If c < 0, then ac > bc.
Division Property
If c > 0, then a  c < b  c.
If c < 0, then a  c > b  c.
Similar statements can be written for a < b, a > b, and a > b.
Any value of a variable that makes an inequality true is a
solution of the inequality.
II. Solve each inequality and graph the
solution on the number line.
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Greater than symbol makes the arrow
point to the right on the # line.
Less than symbol makes the arrow point
to the left on the # line.
If > or < , then shade in the circle. If > or
<, then leave the circle open.
II. Solve each inequality and graph the
solution on the number line.
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Ex 1. 4x – 5 > 13
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Ex 2. 4 – 3p > 16 – p
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Ex 3. 2y + 9 < 5y + 15
Ex. 4 Claire’s test average in her world history class
is 90. The test average is 2/3 of the final grade and
the homework is 1/3 of the final grade. What
homework average does Claire need in order to
have a final grade of at least a 93%?
Final grade = 2/3 (test average) + 1/3 (homework average)
III. Compound Inequalities – is a pair of
inequalities joined by and or or.
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To solve an inequality
involving and, find the
values of the variable
that satisfy both
inequalities. An AND
compound inequality
either has an answer
because the inequalities
INTERSECT or a no
solution answer,
because the inequalities
DON’T INTERSECT.
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To solve an inequality
involving or, find those
values of the variable that
satisfy at least one of
inequalities. An OR
compound inequality either
has an inequality solution
because the inequalities
DON’T INTERSECT or
all real numbers because
the inequalities
INTERSECT and COVER
THE ENTIRE NUMBER
LINE.
III.
Compound Inequalities
Graph the solution of each compound
inequality on a number line.
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Ex 1. 2x + 1 > 3 and 3x – 4 < 17
III. Compound Inequalities
Graph the solution of each compound inequality on
a number line.
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Ex 3. 5x + 1 > 21 or 3x + 2 < -1
III. Compound Inequalities
Graph the solution of each compound inequality on
a number line.
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Ex 4. x + 7 > 4 or x – 2 < 2.
Graph both of the above inequalities.
Writing Activities: Solving Inequalities
11). Which Properties of Inequality differ from the
corresponding Properties of Equality?
Explain and include examples.
12). Why do the graphs of some inequalities include
open circles, while others do not? Explain.
13). Describe two kinds of compound inequalities.