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4.1 Solving Linear
Inequalities
Objective: Solve and graph simple and compound inequalities in
one variable.
What are inequalities?
Inequality Symbols
• Greater than: >
• Less than: <
• Greater than or equal to: ≥
• Less than or equal to: ≤
Inequalities
• Inequalities such as x ≤ 1 and p – 3 > 7 are linear
inequalities in one variable.
• A solution of an inequality in one variable is a value
of the variable that makes the inequality true.
• Example: -4, 0.7, and 1 are solutions of x ≤ 1
• Two inequalities are equivalent if they have the
same solutions
Properties of Inequalities
• To write an equivalent inequality:
• Add the same number to each side.
• Subtract the same number from each side.
• Multiply or divide each side by the same positive
number.
• Multiply or divide each side by the same negative
number and reverse the inequality symbol.
Solve the inequality.
• x – 4 > -6
• -5y + 2 ≥ -13
Solve the inequality.
• 7 – 4x < 1 – 2x
• 2x – 3 > x
Solve
• -x + 3 ≤ -6
• x+3<8
• 3y – 5 < 10
• 2x – 3 > x
Graphing Inequalities
• The graph of an inequality in one variable consists
of all points on a real number line that are solutions
of the inequality.
• To graph an inequality in one variable:
• Use an open dot () for < or >
• Use a solid dot () for ≤ or ≥
Graphing Inequalities
• Graph x < 2
• Graph x ≥ 1
Solve the inequality and then
• 4x + 3 ≤ 6x – 5
• -x + 2 < -3
Compound Inequalities
A compound inequality is two simple
inequalities joined by the word “and” or the
word “or”
AND
All real numbers greater than
or equal to -2 and less than 1
can be written as:
OR
All real numbers less than -1 or
greater than or equal to 2 can
be written as:
-2 ≤ x < 1
Graph:
x < -1 or x ≥ 2
Graph:
Graph the Compound
Inequality
• x < -2 or x ≥ 3
• x ≤ -2 or x > 3
• -2 < x ≤ 3
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