Download 3.4 (Solving Multi-Step Inequalities)

DJB Lesson(key):
Alg 1A
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Alg-1A_Solving Multi-Step Inequalities (3.4)_L1
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3.4 (Solving Multi-Step Inequalities)
Recall:
Solving an inequality is similar to solving an equation:
We isolate the variable by performing inverse operations to undo operations.
When we have isolated the variable, we have found the solution set .
Exception: with inequalities, if we multiply or divide both sides by a
negative number, then we immediately reverse the direction of the inequality.
Let's review:
What happens when we perform operations on both sides of an inequality:
What happens to the
Operation performed to both sides
the Inequality Symbol
Add or Subtract a positive number:
stays the same
Add or Subtract a negative number:
stays the same
Multiply or Divide both sides by a positive number
stays the same
Multiply or Divide both sides by a negative number
it is reversed
First, let's clarify an important point about multiplying (or dividing) by negatives.
Key Concept:
Sometimes we may need to multiply (or divide) by a negative number
as we are simplifying the left or right side of the inequality.
But we only reverse the sign of the inequality when we multiply (or divide)
the entire left side and entire right side by a new negative number that is
not already a part of the inequality.
If
a < b
Then
-a >-b
0
-b < -a
-a > -b
Ex:
Given equality:
This simplifies to:
a<b
<< The direction is reversed
when we have changed the
signs of both sides
(multiplied both sides by -1)
-2(x - 5) < (-3)(4)
-2x + 10 < -12
Here, we have not multiplied both sides by a negative.
We have only re-written each side in a different (simplified) way.
So we do not reverse the direction of the inequality (in this step).
We have not really changed the sign of the entire left & right sides
in this step. We can see however, that we will eventually change the
direction of the inequality symbol when we divide by (-2) as the final step .
Let's apply these basic principles in solving some more challenging
inequality problems such as the following:
(1)
3x - 7 > -22
<< 2-step problem
(2)
-2(2x) - 5x - 3 < 24
<< need to simplify & combine "like terms"
(3)
-3(x - 7) + (x+5) > 21
<< need to 1st apply the Distributive Property
(4)
14x - 7 < 11x - 4
<< the variable is on both sides
To solve these, we use the same steps as for solving equations:
1. Simplify each side (separately):
1st, apply the Distrib. Prop. to elimin. ( )
2nd, combine any Like Terms
2. If the variable appears on both sides,
then eliminate the variable from one side
3. Finish with Two-Step (using inverse operations)
1st, undo Addition or Subtraction
2nd, undo Multiplication or Division ( reversing the inequality
symbol if we multiply or divide both sides by a negative)
4. Check answer by substituting value(s).
Should check 2 values with inequalities
Solve each Inequality. Graph the solution set.
Cornell Notes format: For future practice: cover right side to hide ansser; re-work problem & check.
(1)
3x - 7 > -22
3x - 7
3x - 7 + 7
3x
(1/3)*3x
x
> - 22
> - 22 + 7
> - 15
> (-15)*(1/3)
>-5
<< add 7 to both sides
<< simplify
<< divide both sides by 3
<< simplify
-5 is not a solution
-6
-5
-4
-3
-2
-1
0
1
(so open dot)
2
Check by substituting into original equation (check 2 values)
Check any number > -5
Check -5
3(-4) - 7 > -22
3(-5) - 7 > -22
Simplify to show this is True
Simplify to show this is False
(2)
-2(2x) - 5x - 3 < 24
-2(2x) - 5x - 3 < 24
-4x - 5x - 3 < 24
-9x - 3 < 24
-9x - 3 + 3 < 24 + 3
-9x < 27
(-1/9)*(-9x) > 27*(-1/9)
<< simplify
<< combine Like Terms
<< add 3 to both sides
<< simplify
<< divide both sides by ( -9 )
and reverse inequality sign
x
> -3
<< simplify
-3 is a solution
-6
-5
-4
-3
-2
-1
0
1
2
(so closed dot)
Check by substituting into original equation (check 2 values)
Check any number > -3
Check -3
-2(2*(-2)) -5*(-2) - 3 < 24
(3)
-2(2*(-3)) - 5(-3) - 3 < 24
Simplify to show this is True
Simplify to show this is True
-3(x - 7) + (x+5) > 21
(-3)*x - 3*(-7) + x + 5 > 21
<< Apply Distributive Property
-3(x - 7) + (x+5) > 21
-3x + 21 + x + 5 > 21
-2x + 26 > 21
-2x + 26 - 26 > 21 - 26
-2x > -5
(-1/2)*(-2x) < (-5)*(-1/2)
<< simplify
<< combine Like Terms
<< subtract 26 on both sides
<< simplify
<< divide both sides by ( -2 )
and reverse inequality sign
x
x
< 5/2
< 2.5
<< simplify
2.5 is not a solution
-4
-3
-2
-1
0
1
2
3
4
(so open dot)
Check by substituting into original equation (check 2 values)
Check any number < 2.5
-3(2 - 7) + (2 + 5) > 21
(4)
Check 2.5
-3(2.5-7) + (2.5+5) > 21
Simplify to show this is True
Simplify to show this is False
14x - 7
< 11x - 4
14x - 11x - 7 < 11x - 11x - 4
3x - 7 < - 4
<< eliminate x-term from the right
14x - 7 < 11x - 4
<< simplify
<< combine Like Terms
3x - 7 + 7 < - 4 + 7
3x < 3
(1/3)*(3x) < 3*(1/3)
x < 1
-4
-3
-2
-1
0
1
<< add 7 to both sides
<< simplify
<< divide both sides by 3
<< simplify
2
3
4
1 is a solution
(so closed dot)
Check by substituting into original equation (check 2 values)
Check any number < 1
Check x=1
14(0) - 7 < 11(0) - 4
14(1) - 7 < 11(1) - 4
Simplify to show this is True
Simplify to show this is True