Compare the process of solving |x – 1| + 1 < 15 to that of solving |x – 1| + 1 > 15.

In the first case we'd subtract 1 from both sides, obtaining |x-1|<14.

In the second case we'd also subtract 1 from both sides, and would obtain
|x-1|>14.

What would the graphs look like?

In the first case, the graph would be on the x-axis with "center" at x=1. From this center count 14 units to the right, and then place a circle around that location (which would be at x=15). Next, count 14 units to the left of this center, and place a circle around that location (which would be -13). Draw a line segment connecting the two circles. Notice that all of the solutions are between -13 and +15, not including these endpoints.

In the second case, x has to be greater than 15 or less than -13. Draw an arrow from x=1 to the left, and then draw a separate arrow from 15 to the right. None of the values in between are solutions.

Yogeshkumari Yogeshkumari · Answer 2 · 2022-06-29T14:16:14+02:00

Comparison of the given inequality

|x – 1| + 1 < 15 represents value of x lies between -13 < x < 15 . Intersection of the region lies within the range of (-13, 15 ).

|x – 1| + 1 > 15 represents value of x lies x >15 or x< -13. Intersection of the region is empty.

What is inequality?

" Inequality is defined as the relation between the two quantity using variables with the sign of inequality <, >, ≤, ≥."

According to the question,

Given inequality,

Simplify the given inequality we get,

[tex]|x - 1| + 1 < 15 \\\\\implies |x - 1| +1-1 < 15-1\\\\\implies |x - 1| < 14\\\\\implies -14 < x - 1 < 14\\\\\implies -14+1 < x-1+1 < 14+1\\\\\implies -13 < x < 15[/tex]

From the graph, region lies between -13 < x < 15 in red colour.

Simplify the inequality |x – 1| + 1 > 15 we get,

[tex]|x - 1| + 1 > 15\\\\\implies |x - 1| + 1 -1 > 15-1\\\\\implies |x - 1| > 14\\\\\implies x - 1 > 14 or x-1 < -14\\\\\implies x > 15 or x < -13[/tex]

From graph , region x>15 or x<-13 in blue colour. No common region.

Hence, Comparison of the given inequality

|x – 1| + 1 < 15 represents value of x lies between -13 < x < 15 . Intersection of the region lies within the range of (-13, 15 ).

|x – 1| + 1 > 15 represents value of x lies x >15 or x< -13. Intersection of the region is empty.

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