4.1 Solving Linear Inequalities 11/2/2012. You have learned how to solve equations with 1 variable....
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Transcript of 4.1 Solving Linear Inequalities 11/2/2012. You have learned how to solve equations with 1 variable....
4.1Solving Linear Inequalities
11/2/2012
You have learned how to solve equations with 1 variable.Ex. x + 3 = 7 -3 -3 x = 4
Ex. x - 5 = 2 +5 +5 x = 7
Ex. 3x = 12 3 3 x = 4
Ex. y = 6 2 y = 12
·22·
To solve inequality in 1 variable
All the rules apply for solving equations in 1 variable except when dividing or multiplying both sides by a negative number. Rule: When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.
Example 1 Inequality with a Variable on One Side
Solve the inequality.
a. b.>x 4– 6– – ≥5y 2 13–+
SOLUTION
a. >x 4– 6– Write original inequality.
>x 2– Add 4 to each side.
ANSWER
The solution is all real numbers greater than 2.–
+ 4 + 4
Example 1 Inequality with a Variable on One Side
b. – ≥5y 2 13–+ Write original inequality.
– ≥5y 15– Subtract 2 from each side.
ANSWER
The solution is all real numbers less than or equal to 3.
Divide each side by 5 and reverse the inequality.
– y ≤ 3
-2 -2
-5 -5
Graphing the solution of an inequality in 1 variable.
For > or <, doughnut ( )For ≤ or ≥, solid circle ( )
Then shade the side of the number line according to which way the inequality symbol is pointing when the variable is on the left side
Ex. x > -1
Ex. x ≤ 3
Example 2 Inequality with a Variable on Both Sides
Solve . Graph the solution.
Add 2x to each side.<7 2x– 1
<7 4x 1 2x––
SOLUTION
Write original inequality.<7 4x 1 2x––
<2x– 6– Subtract 7 from each side.
>x 3 Divide each side by 2 and reverse the inequality.
–
+2x +2x
-2 -2
Example 2 Inequality with a Variable on Both Sides
ANSWER
The solution is all real numbers greater than 3. The graph is shown at the bottom.
2 3 4 511– 0
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Inequality
<x 3 8+1.
5≤4 x–2.
ANSWER <x 5
210 3 4 5 6
ANSWER ≥x 1–
0 22–
Checkpoint
Solve the inequality. Then graph your solution.
– x>2x 34.
Solve an Inequality
2>2x 1–3.
3x >ANSWER
210 3 4 5 6
x >2
3ANSWER
1 20
Example 3 Use a Simple Inequality
Amusement Park Admission to an amusement park costs $9 and each ride ticket costs $1.50. The total amount A in dollars spent is given by where t is the number of ride tickets. Use an inequality to describe the number of ride tickets you can buy if you have at most $45 to spend at the park.
=A 1.5t + 9
SOLUTION The most money you can spend is $45.A 45≤
Substitute for A.45≤1.5t + 9 1.5t + 9
Subtract 9 from each side.36≤1.5t
Divide each side by 1.5.24≤t
ANSWER You can buy up to 24 ride tickets.
COMPOUND INEQUALITY2 simple inequalities joined by the word “and” or the word “or”. Ex: ANDAll real numbers greater than or equal to -2 AND less than 1.
Ex: ORAll real numbers less than -1 OR greater than or equal to 2
-2≤ x < 1
x < -1 or x ≥ 2
Example 4 Solve an “Or” Compound Inequality
Solve or3x + 2 8< 2x 9 3.– >
SOLUTION
Solve each part separately.
FIRST INEQUALITY SECOND INEQUALITY
3x + 2 8< 2x 9 3– >Write first inequality.
Write second inequality.
3x 6< 2x 12>Subtract 2 from each side.
Add 9 to eachside.
x 2< x 6>Divide each sideby 3.
Divide each side by 2.
Checkpoint
Solve the inequality. Then graph your solution.
Solve Compound Inequalities
5. 4 x< + 5 7<
6. 3x≤ + 81– 8≤
ANSWER x 2<1 <–
2– 20
ANSWER x≤3– 0≤
4– 2– 0..
Checkpoint
Solve the inequality. Then graph your solution.
Solve Compound Inequalities
7. x + 3 4 or ≤ x 6– ≥ 1–
ANSWER x 1 or≤ x 5≥
420. .
8. 2x 6 0<– –x 4 or– >
ANSWER x 4 or–< x 3–>
06– 4– 2–
Homework4.1 p.175 #14-22 even, 23-28all, 30-38even, 40-44all, 50-56even