Graph 3|3-x|+1<5

Math
3|3-x|+1<5
Write 3|3-x|+1<5 as a piecewise.
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To find the interval for the first piece, find where the inside of the absolute value is non-negative.
3-x≥0
Solve the inequality.
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Subtract 3 from both sides of the inequality.
-x≥-3
Multiply each term in -x≥-3 by -1
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Multiply each term in -x≥-3 by -1. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
(-x)⋅-1≤(-3)⋅-1
Multiply (-x)⋅-1.
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Multiply -1 by -1.
1x≤(-3)⋅-1
Multiply x by 1.
x≤(-3)⋅-1
x≤(-3)⋅-1
Multiply -3 by -1.
x≤3
x≤3
x≤3
In the piece where 3-x is non-negative, remove the absolute value.
3(3-x)+1<5
To find the interval for the second piece, find where the inside of the absolute value is negative.
3-x<0
Solve the inequality.
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Subtract 3 from both sides of the inequality.
-x<-3
Multiply each term in -x<-3 by -1
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Multiply each term in -x<-3 by -1. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
(-x)⋅-1>(-3)⋅-1
Multiply (-x)⋅-1.
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Multiply -1 by -1.
1x>(-3)⋅-1
Multiply x by 1.
x>(-3)⋅-1
x>(-3)⋅-1
Multiply -3 by -1.
x>3
x>3
x>3
In the piece where 3-x is negative, remove the absolute value and multiply by -1.
3(-(3-x))+1<5
Write as a piecewise.
{3(3-x)+1<5x≤33(-(3-x))+1<5x>3
Simplify 3(3-x)+1<5.
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Simplify each term.
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Apply the distributive property.
{3⋅3+3(-x)+1<5x≤33(-(3-x))+1<5x>3
Multiply 3 by 3.
{9+3(-x)+1<5x≤33(-(3-x))+1<5x>3
Multiply -1 by 3.
{9-3x+1<5x≤33(-(3-x))+1<5x>3
{9-3x+1<5x≤33(-(3-x))+1<5x>3
Add 9 and 1.
{-3x+10<5x≤33(-(3-x))+1<5x>3
{-3x+10<5x≤33(-(3-x))+1<5x>3
Simplify 3(-(3-x))+1<5.
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Simplify each term.
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Apply the distributive property.
{-3x+10<5x≤33(-1⋅3–x)+1<5x>3
Multiply -1 by 3.
{-3x+10<5x≤33(-3–x)+1<5x>3
Multiply –x.
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Multiply -1 by -1.
{-3x+10<5x≤33(-3+1x)+1<5x>3
Multiply x by 1.
{-3x+10<5x≤33(-3+x)+1<5x>3
{-3x+10<5x≤33(-3+x)+1<5x>3
Apply the distributive property.
{-3x+10<5x≤33⋅-3+3x+1<5x>3
Multiply 3 by -3.
{-3x+10<5x≤3-9+3x+1<5x>3
{-3x+10<5x≤3-9+3x+1<5x>3
Add -9 and 1.
{-3x+10<5x≤33x-8<5x>3
{-3x+10<5x≤33x-8<5x>3
{-3x+10<5x≤33x-8<5x>3
Solve -3x+10<5 when x≤3.
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Solve -3x+10<5 for x.
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Move all terms not containing x to the right side of the inequality.
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Subtract 10 from both sides of the inequality.
-3x<5-10
Subtract 10 from 5.
-3x<-5
-3x<-5
Divide each term by -3 and simplify.
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Divide each term in -3x<-5 by -3. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
-3x-3>-5-3
Cancel the common factor of -3.
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Cancel the common factor.
-3x-3>-5-3
Divide x by 1.
x>-5-3
x>-5-3
Dividing two negative values results in a positive value.
x>53
x>53
x>53
Find the intersection of x>53 and x≤3.
53<x≤3
53<x≤3
Solve 3x-8<5 when x>3.
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Solve 3x-8<5 for x.
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Move all terms not containing x to the right side of the inequality.
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Add 8 to both sides of the inequality.
3x<5+8
Add 5 and 8.
3x<13
3x<13
Divide each term by 3 and simplify.
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Divide each term in 3x<13 by 3.
3×3<133
Cancel the common factor of 3.
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Cancel the common factor.
3×3<133
Divide x by 1.
x<133
x<133
x<133
x<133
Find the intersection of x<133 and x>3.
3<x<133
3<x<133
Find the union of the solutions.
53<x<133
<div data-graph-input="{"graphs":[{"ascii":"(5)/(3)<x
Graph 3|3-x|+1<5

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