chapter 1.7 inequalities. an inequality says that one expression is greater than, or greater than or...
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Chapter 1.7
Inequalities
An inequality says that one expression is greater than, or greater than or equal to, or less than, or less than or equal to, another.
As with equations, a value of the variable for which the inequality is true is a solution of the inequality.
Two inequalities with the same solution set are equivalent.
Properties of Inequality
For real numbers a, b, and c:
If a<b, then a + c < b + c,
If a<b, and if c> 0 then ac < bc,
If a<b, and if c< 0 then ac > bc.
Properties of Inequality
Multiplication may be replaced by division in properties 2 and 3.
Always remember to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.
Linear Inequalities in One Variable
A linear inequality in one variable is an inequality that can be written in the form
ax + b > 0
0.awith
numbers real are b and a where
used) be alsomay or
, symbols theof(Any
Solve -3x + 5 > -7
Example 1 Solving a Linear Inequality
The original inequality is satisfied by any real number less than 4. The solution set can be written {x| x < 4}.
A graph of the solution set is shown in Figure 9, where the parenthesis is used to show that 4 itself does not belong to the solution set.
Open Interval
,a
axx
(a
Open Interval
ba
bxax
,
(a
)b
Open Interval
b
bxx
,
)b
Half-open Interval
,a
axx
[a
Half-opened Interval
ba
bxax
,
(a
]b
Half-opened Interval
ba
bxax
,
[a
)b
Half-open Interval
b
bxx
,
]b
Closed Interval
ba
bxax
,
[a
]b
All real numbers
,
realisxx
Example 2 Solving a Linear Inequality
2x. 73x - 4 Solve
Give the solution set in interval notation and graph it.
Interval Notation
[0-1 54321-5 -4 -3 -2
5
3x
Three-Part Inequalities
The inequality -2 < 5 +3x < 20 is the next example that 5 + 3x is between -2 and 20.
This inequality is solved using an extension of the properties of inequality given earlier, working with all three expressions at the same time.
Example 3 Solving a Three-Part Inequality
Solve -2 < 5 +3x < 20
Graph
0-1 54321-5 -4 -3 -2
53
7 x
A product will break even, or begin to produce a profit, only if the revenue from selling the product at least equals the cost of producing it. If R represents revenue and C is cost, then the break-even point is the point where R = C.
If the revenue and cost of a certain product are given by
R = 4x and C = 2x + 1000
where x is the number of units produced and sold, at what production level does R at least equal C?
Example 4 Finding the Break-Eve Point
If the revenue and cost of a certain product are given by
R = 4x and C = 2x + 1000
where x is the number of units produced and sold, at what production level does R at least equal C?
Example 4 Finding the Break-Eve Point
for x. solve and C RSet
Quadratic Inequality
A quadratic inequality is an inequality that can be written in the form
ax2 + bx + c < 0
0.awith
c and b, a, numbers realfor
used) be alsomay or
, symbols theof(Any
Example 5 Solving a Quadratic Inequality
Solve x2 – x - 12
y
x
Example 6 Solving a Quadratic Inequality
0 12 -5x 2x Solve 2
y
x
If a projectile is launched from ground with an initial velocity of 96 ft per sec. its height in feet t seconds after launching is s feet, where
s = -16t2 + 96 t
When will the projectile be greater than 80 ft above ground level.
Example 7 Solving a Problem Involving the Height of a Projectile
Rational Inequalities
Inequalities involving rational expressions such as
are called rational inequalities, and are solved in a manner similar to the procedure for solving quadratic inequalities.
Example 8 Solving a Rational Inequality
14x
5 Solve
y
x
Example 9 Solving a Rational Inequality
543x
1-2x Solve
y
x
Example 9 Solving a Rational Inequality
543x
1-2x Solve
y
x
Homework
Section 1.7 # 1 - 52