Solving Inequalities

1
Chapter 1

• Section 4
• Solving Inequalities

2
Solving Inequalities
ALGEBRA 2 LESSON 1-4
(For help, go to Lessons 1-1 and 1-3.)
State whether each inequality is true or false.
Solve each equation.
3
Solving Inequalities
ALGEBRA 2 LESSON 1-4
Solutions
2. 5 lt 12, false 4. 5 12, false 6. 5 5,
true 8. 5x 9(x 8) 12 5x 9x 72 124x
60x 15
1. 5 lt 12, true 3. 5 12, false 5. 5 5,
true 7. 3x 3 2x 3 3x 2x 3 3x 6
4
Inequalities

• The solutions include more than one number
• Ex 2 lt x values that x could be include 3, 7,
  45
• All of the rules for solving equations apply to
  inequalities, with one added
• If you multiply or divide by a NEGATIVE you must
  FLIP the sign. (lt becomes gt and gt becomes lt)
• When graphing on a number line
• Open dot for lt or gt
• Closed (solid) dot for or
• The shading should be easy to see (a slightly
  elevated line is ok) --- see examples

5
Solving Inequalities
ALGEBRA 2 LESSON 1-4
Solve 2x lt 3(x 5). Graph the solution.
2x lt 3(x 5)
2x lt 3x 15 Distributive Property
5x lt 15 Subtract 3x from both sides.
x gt 3 Divide each side by 5 and reverse
the inequality.
6
Try These Problems

• Solve each inequality. Graph the solution.
• 3x 6 lt 27
• 3x lt 33 x lt 11
• 12 2(3n 1) 22
• 12 6n 2 2212 6n 24-12 6n-2 n

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Solve 7x 7(2 x). Graph the solution.
7x 7(2 x)
7x 14 7x Distributive Property
0 14 Subtract 7x from both sides.
The last inequality is always false, so 7x 7(2
x) is always false. It has no solution.
8
Try These Problems

• Solve. Graph the solution.
• 2x lt 2(x 1) 3
• 2x lt 2x 2 32x lt 2x 5 0 lt 5All Real
  Numbers
• 4(x 3) 7 4x 1
• 4x 12 7 4x 1 4x 12 4x 1
  12 1 No Solution

9
Solving Inequalities
A real estate agent earns a salary of 2000 per
month plus 4 of the sales. What must the sales
be if the salesperson is to have a monthly income
of at least 5000?
The sales must be greater than or equal to
75,000.
10
Try This Problem

• A salesperson earns a salary of 700 per month
  plus 2 of the sales. What must the sales be if
  the salesperson is to have a monthly income of at
  least 1800?
• 700 .02x 1800
• .02x 1100
• x 55000
• The sales must be at least 55,000.

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Compound Inequalities

• Compound Inequality a pair of inequalities
  joined by and or or
• Ex -1 lt x and x 3 which can be written
  as -1 lt x 3
• x lt -1 or x 3
• For and statements the value must satisfy both
  inequalities
• For or statements the value must satisfy one of
  the inequalities

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And Inequalities

• Graph the solution of 3x 1 gt -28 and 2x 7 lt
  19.
• 3x gt -27 and 2x lt 12
• x gt -9 and x lt 6
• Graph the solution of -8 lt 3x 1 lt19
• -9 lt 3x lt 18
• -3 lt x lt 6

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Or Inequalities
ALGEBRA 2 LESSON 1-4
Graph the solution of 3x 9 lt 3 or 2x 1 lt 5.
3x 9 lt 3 or 2x 1 lt 5
x lt 4 or x gt 2
14
Try These Problems

1. Graph the solution of 2x gt x 6 and x 7 lt 2
2. x gt 6 and x lt 9
3. Graph the solution of x 1 lt 3 or x 3 gt 8
4. x lt 4 or x gt 11

15
Homework

• Practice 1.4 All omit 23