Chapter 1.7 Inequalities. An inequality says that one expression is greater than, or greater than or...
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![Page 1: 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,](https://reader036.fdocuments.us/reader036/viewer/2022062321/56649f0e5503460f94c22caa/html5/thumbnails/1.jpg)
Chapter 1.7
Inequalities
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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.
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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.
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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.
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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
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Solve -3x + 5 > -7
Example 1 Solving a Linear Inequality
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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.
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Open Interval
,a
axx
(a
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Open Interval
ba
bxax
,
(a
)b
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Open Interval
b
bxx
,
)b
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Half-open Interval
,a
axx
[a
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Half-opened Interval
ba
bxax
,
(a
]b
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Half-opened Interval
ba
bxax
,
[a
)b
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Half-open Interval
b
bxx
,
]b
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Closed Interval
ba
bxax
,
[a
]b
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All real numbers
,
realisxx
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Example 2 Solving a Linear Inequality
2x. 73x - 4 Solve
Give the solution set in interval notation and graph it.
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Interval Notation
[0-1 54321-5 -4 -3 -2
5
3x
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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.
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Example 3 Solving a Three-Part Inequality
Solve -2 < 5 +3x < 20
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Graph
0-1 54321-5 -4 -3 -2
53
7 x
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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.
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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
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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
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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
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Example 5 Solving a Quadratic Inequality
Solve x2 – x - 12
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y
x
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Example 6 Solving a Quadratic Inequality
0 12 -5x 2x Solve 2
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y
x
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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
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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.
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Example 8 Solving a Rational Inequality
14x
5 Solve
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y
x
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Example 9 Solving a Rational Inequality
543x
1-2x Solve
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y
x
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Example 9 Solving a Rational Inequality
543x
1-2x Solve
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y
x
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Homework
Section 1.7 # 1 - 52