Transcript Ch2-Section 2.8

§ 2.8
Solving Linear
Inequalities
Linear Inequalities in One Variable
A linear inequality in one variable is an inequality
that can be written in the form
ax + b < c
where a, b, and c are real numbers and a is not 0.
This definition and all other definitions, properties and
steps in this section also hold true for the inequality
symbols >, , or .
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solutions to Linear Inequalities
Graphing Solutions to Linear Inequalities
• Use a number line.
• Use a closed circle at the endpoint of an interval if
you want to include the point.
• Use an open circle at the endpoint if you DO NOT
want to include the point.
Represents the set {xx  7}.
7
Represents the set {xx > – 4}.
-4
Interval notation, is used to write solution sets of inequalities.
• Use a parenthesis if you want to include the number
• Use a bracket if you DO NOT want to include the number..
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solutions to Linear Inequalities
x<3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Interval notation: (–∞, 3)
NOT included
–2 < x < 0
-5
-4
-3
-2
-1
0
1
2
3
4
5
3
4
5
Interval notation: (–2, 0)
–1.5  x  3
Included
-5
-4
-3
-2
-1
0
1
2
Interval notation: [–1.5, 3)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Properties of Inequality
Addition Property of Inequality
If a, b, and c are real numbers, then
a < b and a + c < b + c
are equivalent inequalities.
Multiplication Property of Inequality
1. If a, b, and c are real numbers, and c is positive, then
a < b and ac < bc are equivalent inequalities.
2. If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc are equivalent inequalities.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solving Linear Inequalities
Solving Linear Inequalities in One Variable
1) Clear the inequality of fractions by multiplying both sides
by the LCD of all fractions of the inequality.
2) Remove grouping symbols by using the distributive
property.
3) Simplify each side of equation by combining like terms.
4) Write the inequality with variable terms on one side and
numbers on the other side by using the addition property of
equality.
5) Get the variable alone by using the multiplication property
of equality.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solving Linear Inequalities
Example: Solve the inequality. Graph the solution
and give your answer in interval notation.
2(x – 3) < 4x + 10
2x – 6 < 4x + 10
Distribute.
– 6 < 2x + 10
Subtract 2x from both sides.
– 16 < 2x
Subtract 10 from both sides.
– 8 < x or x > – 8
Divide both sides by 2.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
(–8, ∞)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solving Linear Inequalities
Example: Solve the inequality. Graph the solution
and give your answer in interval notation.
x+5x–2
xx–7
Subtract 5 from both sides.
0–7
Subtract x from both sides.
(Always true!)
Since 0 is always greater than –7, the solution is all real
numbers. (Any value we put in for x in the original
statement will give us a true inequality.)
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
(–∞ ∞)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solving Linear Inequalities
Example: Solve the inequality. Give your answer in
interval notation.
a.) 9 < z + 5 < 13
b.) –7 < 2p – 3 ≤ 5
a.) 9 < z + 5 < 13
4<z<8
(4, 8)
b.) –7 < 2p – 3 ≤ 5
–4 < 2p ≤ 8
–2 < p ≤ 4
Subtract 5 from all three parts.
Add 3 to all three parts.
Divide all three parts by 2.
(–2, 4]
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solving Linear Inequalities
Example: Solve the inequality. Give your answer in
graph form.
3x + 9  5(x – 1)
3x + 9  5x – 5
3x – 3x + 9  5x – 3x – 5
Use distributive property on right side.
Subtract 3x from both sides.
9  2x – 5
Simplify both sides.
9 + 5  2x – 5 + 5
Add 5 to both sides.
14  2x
Simplify both sides.
7x
Divide both sides by 2.
Solution:
]
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Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Compound Inequalities
A compound inequality contains two inequality
symbols.
0  4(5 – x) < 8
This means 0  4(5 – x) and 4(5 – x) < 8.
To solve the compound inequality, perform operations
simultaneously to all three parts of the inequality (left,
middle and right).
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Solving Compound Inequalities
Example: Solve the inequality. Give your answer in
interval notation.
0  20 – 4x < 8
0  20 – 4x < 8
0 – 20  20 – 20 – 4x < 8 – 20
Use the distributive property.
Subtract 20 from each part.
– 20  – 4x < – 12
Simplify each part.
5x>3
Divide each part by –4.
Remember that the sign changes direction when you divide by a
negative number.
The solution is (3,5].
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Inequality Applications
Example: Six times a number, decreased by 2, is at least
10. Find the number.
1.) UNDERSTAND
Let x = the unknown number.
“Six times a number” translates to 6x,
“decreased by 2” translates to 6x – 2,
“is at least 10” translates ≥ 10.
Continued
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Finding an Unknown Number
Example continued:
2.) TRANSLATE
Six times
a number
decreased
by 2
is at least
10
6x
–
2
≥
10
Continued
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Finding an Unknown Number
Example continued:
3.) SOLVE
6x – 2 ≥ 10
6x ≥ 12
x≥2
Add 2 to both sides.
Divide both sides by 6.
4.) INTERPRET
Check: Replace “number” in the original statement of the
problem with a number that is 2 or greater.
Six times 2, decreased by 2, is at least 10
6(2) – 2 ≥ 10
10 ≥ 10
State: The number is 2.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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