x-2x+3≥0

Find all the values where the expression switches from negative to positive by setting each factor equal to 0 and solving.

x-2=0

x+3=0

Add 2 to both sides of the equation.

x=2

Subtract 3 from both sides of the equation.

x=-3

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

x=2

x=-3

Consolidate the solutions.

x=2,-3

Set the denominator in x-2x+3 equal to 0 to find where the expression is undefined.

x+3=0

Subtract 3 from both sides of the equation.

x=-3

The domain is all values of x that make the expression defined.

(-∞,-3)∪(-3,∞)

(-∞,-3)∪(-3,∞)

Use each root to create test intervals.

x<-3

-3<x<2

x>2

Test a value on the interval x<-3 to see if it makes the inequality true.

Choose a value on the interval x<-3 and see if this value makes the original inequality true.

x=-6

Replace x with -6 in the original inequality.

(-6)-2(-6)+3≥0

The left side 2.6‾ is greater than the right side 0, which means that the given statement is always true.

True

True

Test a value on the interval -3<x<2 to see if it makes the inequality true.

Choose a value on the interval -3<x<2 and see if this value makes the original inequality true.

x=0

Replace x with 0 in the original inequality.

(0)-2(0)+3≥0

The left side -0.6‾ is less than the right side 0, which means that the given statement is false.

False

False

Test a value on the interval x>2 to see if it makes the inequality true.

Choose a value on the interval x>2 and see if this value makes the original inequality true.

x=4

Replace x with 4 in the original inequality.

(4)-2(4)+3≥0

The left side 0.285714‾ is greater than the right side 0, which means that the given statement is always true.

True

True

Compare the intervals to determine which ones satisfy the original inequality.

x<-3 True

-3<x<2 False

x>2 True

x<-3 True

-3<x<2 False

x>2 True

The solution consists of all of the true intervals.

x<-3 or x≥2

The result can be shown in multiple forms.

Inequality Form:

x<-3 or x≥2

Interval Notation:

(-∞,-3)∪[2,∞)

<div data-graph-input="{"graphs":[{"ascii":"x=2","color":0,"isGrey":false,"dashed":false,"holes":[]}],"asymptotes":[],"segments":[],"areaBetweenCurves":[],"points":[],"HasGraphInput":true}” class=”GraphWrapper”>