|10-4x|<13

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

10-4x≥0

Solve the inequality.

Subtract 10 from both sides of the inequality.

-4x≥-10

Divide each term by -4 and simplify.

Divide each term in -4x≥-10 by -4. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-4x-4≤-10-4

Cancel the common factor of -4.

Cancel the common factor.

-4x-4≤-10-4

Divide x by 1.

x≤-10-4

x≤-10-4

Cancel the common factor of -10 and -4.

Factor -2 out of -10.

x≤-2(5)-4

Cancel the common factors.

Factor -2 out of -4.

x≤-2⋅5-2⋅2

Cancel the common factor.

x≤-2⋅5-2⋅2

Rewrite the expression.

x≤52

x≤52

x≤52

x≤52

x≤52

In the piece where 10-4x is non-negative, remove the absolute value.

10-4x<13

To find the interval for the second piece, find where the inside of the absolute value is negative.

10-4x<0

Solve the inequality.

Subtract 10 from both sides of the inequality.

-4x<-10

Divide each term by -4 and simplify.

Divide each term in -4x<-10 by -4. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-4x-4>-10-4

Cancel the common factor of -4.

Cancel the common factor.

-4x-4>-10-4

Divide x by 1.

x>-10-4

x>-10-4

Cancel the common factor of -10 and -4.

Factor -2 out of -10.

x>-2(5)-4

Cancel the common factors.

Factor -2 out of -4.

x>-2⋅5-2⋅2

Cancel the common factor.

x>-2⋅5-2⋅2

Rewrite the expression.

x>52

x>52

x>52

x>52

x>52

In the piece where 10-4x is negative, remove the absolute value and multiply by -1.

-(10-4x)<13

Write as a piecewise.

{10-4x<13x≤52-(10-4x)<13x>52

Simplify -(10-4x)<13.

Apply the distributive property.

{10-4x<13x≤52-1⋅10-(-4x)<13x>52

Multiply -1 by 10.

{10-4x<13x≤52-10-(-4x)<13x>52

Multiply -4 by -1.

{10-4x<13x≤52-10+4x<13x>52

{10-4x<13x≤52-10+4x<13x>52

{10-4x<13x≤52-10+4x<13x>52

Solve 10-4x<13 for x.

Move all terms not containing x to the right side of the inequality.

Subtract 10 from both sides of the inequality.

-4x<13-10

Subtract 10 from 13.

-4x<3

-4x<3

Divide each term by -4 and simplify.

Divide each term in -4x<3 by -4. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-4x-4>3-4

Cancel the common factor of -4.

Cancel the common factor.

-4x-4>3-4

Divide x by 1.

x>3-4

x>3-4

Move the negative in front of the fraction.

x>-34

x>-34

x>-34

Find the intersection of x>-34 and x≤52.

-34<x≤52

-34<x≤52

Solve -10+4x<13 for x.

Move all terms not containing x to the right side of the inequality.

Add 10 to both sides of the inequality.

4x<13+10

Add 13 and 10.

4x<23

4x<23

Divide each term by 4 and simplify.

Divide each term in 4x<23 by 4.

4×4<234

Cancel the common factor of 4.

Cancel the common factor.

4×4<234

Divide x by 1.

x<234

x<234

x<234

x<234

Find the intersection of x<234 and x>52.

52<x<234

52<x<234

Find the union of the solutions.

-34<x<234

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