Solve absolute value inequality. |x−1|≥2

Question

Solve absolute value inequality.  |x−1|≥2

scoollato7o · Accepted Answer

Step 1  the given inequality is:  |x−1|≥2  we have to solve the given inequality.  Step 2  the given inequality is |x−1|≥2  as we know that if |x|≥a  then x≤−a or x≥a  that implies x∈(−∞,−a]∪[a,∞)  therefore,  |x−1|≥2  then  x−1≤−2 or x−1≥2  x−1≤−2  x≤−2+1  x≤−1  or  x−1≥2  x≥2+1  x≥3  Step 3  therefore the solution of the given inequality is x≤−1 or x≥3  that implies the solution of the given inequality is x∈(−∞,−1]∪[3,∞)

Donald Cheek · Accepted Answer

Step 1
The absolute value inequality is given as,

| x - 1 | \geq 2

If an absolute inequality is given as,

| f (x) | \geq a

,

| f (x) | \geq a

= {\begin{cases} f (x) \geq a & f (x) > 0 \\ f (x) \leq - a & f (x) < 0 \end{cases}

Step 2
On solving the inequality, we get

x - 1 \geq 2

x \geq 2 + 1

x \geq 3

...(i)
And,

x - 1 \leq - 2

x \leq - 2 + 1

x \leq - 1

...(ii)
Therefore, the interval for the solution of the given inequality is

x \in (- \infty, - 1] \cup [3, \infty)

Solve absolute value inequality. |x-1|\geq 2

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