Graph 2/(x-4)<1

Math
2x-4<1
Move 1 to the left side of the equation by subtracting it from both sides.
2x-4-1<0
Simplify 2x-4-1.
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To write -1 as a fraction with a common denominator, multiply by x-4x-4.
2x-4-1⋅x-4x-4<0
Simplify terms.
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Combine -1 and x-4x-4.
2x-4+-(x-4)x-4<0
Combine the numerators over the common denominator.
2-(x-4)x-4<0
2-(x-4)x-4<0
Simplify the numerator.
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Apply the distributive property.
2-x–4x-4<0
Multiply -1 by -4.
2-x+4x-4<0
Add 2 and 4.
-x+6x-4<0
-x+6x-4<0
Simplify with factoring out.
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Factor -1 out of -x.
-(x)+6x-4<0
Rewrite 6 as -1(-6).
-(x)-1(-6)x-4<0
Factor -1 out of -(x)-1(-6).
-(x-6)x-4<0
Simplify the expression.
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Rewrite -(x-6) as -1(x-6).
-1(x-6)x-4<0
Move the negative in front of the fraction.
-x-6x-4<0
-x-6x-4<0
-x-6x-4<0
-x-6x-4<0
Find all the values where the expression switches from negative to positive by setting each factor equal to 0 and solving.
x-6=0
x-4=0
Add 6 to both sides of the equation.
x=6
Add 4 to both sides of the equation.
x=4
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
x=6
x=4
Consolidate the solutions.
x=6,4
Find the domain of -x-6x-4.
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Set the denominator in x-6x-4 equal to 0 to find where the expression is undefined.
x-4=0
Add 4 to both sides of the equation.
x=4
The domain is all values of x that make the expression defined.
(-∞,4)∪(4,∞)
(-∞,4)∪(4,∞)
Use each root to create test intervals.
x<4
4<x<6
x>6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Test a value on the interval x<4 to see if it makes the inequality true.
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Choose a value on the interval x<4 and see if this value makes the original inequality true.
x=0
Replace x with 0 in the original inequality.
2(0)-4<1
The left side -0.5 is less than the right side 1, which means that the given statement is always true.
True
True
Test a value on the interval 4<x<6 to see if it makes the inequality true.
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Choose a value on the interval 4<x<6 and see if this value makes the original inequality true.
x=5
Replace x with 5 in the original inequality.
2(5)-4<1
The left side 2 is not less than the right side 1, which means that the given statement is false.
False
False
Test a value on the interval x>6 to see if it makes the inequality true.
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Choose a value on the interval x>6 and see if this value makes the original inequality true.
x=8
Replace x with 8 in the original inequality.
2(8)-4<1
The left side 0.5 is less than the right side 1, which means that the given statement is always true.
True
True
Compare the intervals to determine which ones satisfy the original inequality.
x<4 True
4<x<6 False
x>6 True
x<4 True
4<x<6 False
x>6 True
The solution consists of all of the true intervals.
x<4 or x>6
<div data-graph-input="{"graphs":[{"ascii":"x6","color":0,"isGrey":false,"dashed":false,"holes":[]}],"asymptotes":[],"segments":[],"areaBetweenCurves":[],"points":[],"HasGraphInput":true}” class=”GraphWrapper”>
Graph 2/(x-4)<1

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