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# algebra solving absolute value equations and inequalities

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• 1. Ix+3I=4
Ix+3I≥4
Solve absolute equalities
and inequalities
Ix+3I<4
• 2. IxI=4
x=4 ,x=-4
X=4,-4
Now GRAPH….
• 3. Graph IxI=4
Points (-4,4)
-4
4
0
• 4. Ix+3I=4
x+3=4 ,x+3=-4
X+3=-4
-3 -3
-7
x=-7
X+3=4
-3 -3
1
x=1
X=-7,1
• Now CHECK….
• Ix+3I=4
x+3=4 ,x+3=-4
Plug in 1 for x.
1+3=4
4=4
Plug in -7 for x.
-7+3=-4
-4=-4
X=-7,1
• Now GRAPH….
• Graph Ix+3I=4
Points (-7,1)
-7 1
0
• 5. Ix+5I>4
x+5>4 , x+5<-4
-5 -5
-1
X>-1
-5 -5
-9
X<-9
x<-9,X>-1 or (-∞,-9)U(-1,∞)
• Now CHECK….
• Ix+5I>4
x+5>4 , x+5<-4
Plug in anything
less than -9 for x.
Like -10.
-10+5<-4
-5<-4
Plug in anything
greater than -1 for x.
Like 0.
0+5>4
5>4
x<-9,X>-1 or (-∞,-9)U(-1,∞)
• Now GRAPH….
• Graph Ix+5I>4
Open circles
Because its
Less than and
Greater than.
Not equal too.
Points (-9,-1)
x<-9 x>-1
0
• 6. Ix+5I≥4
x+5≥4 , x+5≤-4
-5 -5
-1
X≥-1
-5 -5
-9
X≤-9
x≤-9,X≥-1 or (-∞,-9]U[-1,∞)
• Now CHECK….
• Ix+5I≥4
x+5≥4 , x+5≤-4
Plug in anything
less than -9 for x.
Like -10.
-10+5<-4
-5<-4
Plug in anything
greater than -1 for x.
Like 0.
0+5>4
5>4
x<-9,X>-1 or (-∞,-9]U[-1,∞)
• Now GRAPH….
• Graph Ix+5I≥4
Closed circles
Because its
Less than, equal
too andGreater
Than, equal too.
Points (-9,-1)
x≤-9 x≥-1
0
• 7. I2xI≥4
2x≥4 , 2x≤-4
2 2
X≥2
2 2
X≤-2
x≤-2,X≥2 or (-∞,-2]U[-2, ∞)
• Now CHECK….
• I2xI≥4
2x≥4 , 2x≤-4
Plug in anything
less than -2 for x.
Like -3.
2(-3)≤-6
-6≤-4
Plug in anything
greater than 2 for x.
Like 3.
2(3)≥4
6≥4
x≤-2,X≥2 or (-∞,-2]U[2, ∞)
• Now GRAPH….
• Graph I2xI≥4
Closed circles
Because its
Less than, equal
too andGreater
Than, equal too.
Points (-2,2)
x≤-2 x≥2
0
• 8. I-2xI≥4
-2x≥4 , -2x≤-4
-2 -2
X≥2
-2 -2
Now when you divide by a negative
Number, you HAVE to flip the sign.
X≤-2
x≤-2,x≥2 or (-∞,-2]U[-2, ∞)
• Now CHECK….
• I-2xI≥4
-2x≥4 , -2x≤-4
Plug in anything
greater than 2 for x.
Like 3.
-2(3)≤-4
-6≤-4
Plug in anything
less than -2 for x.
Like -3.
-2(-3)≥6
6≥4
x≤-2,X≥2 or (-∞,-2]U[2, ∞)
• Now GRAPH….
• Graph I-2xI≥4
Closed circles
Because its
Less than, equal
too andGreater
Than, equal too.
Points (-2,2)
x≤-2 x≥2
0