Advertisements
Advertisements
Question
Solve for x, |x + 1| + |x| > 3.
Solution
On L.H.S. of the given inequality, We have two terms both containing modulus. By equating the expression within the modulus to zero, we get x = –1, 0 as critical points. These critical points divide the real line in three parts as (– `oo`, – 1), [–1, 0), [0, `oo`).
Case I: When `–oo < x < –1`
|x + 1| + |x| > 3
⇒ –x – 1 – x > 3
⇒ x < –2
Case II: When –1 ≤ x < 0,
|x + 1| + |x| > 3
⇒ x + 1 – x > 3
⇒ 1 > 3 ....(Not possible)
Case III: When 0 ≤ x < `oo`,
|x + 1| + |x| > 3
⇒ x + 1 + x > 3
⇒ x > 1
Combining the results of cases (I), (II) and (III), We get x ∈ `(–oo , –2) ∪ (1, oo)`.
APPEARS IN
RELATED QUESTIONS
Solve: −4x > 30, when x ∈ R
Solve: −4x > 30, when x ∈ Z
\[\frac{3x - 2}{5} \leq \frac{4x - 3}{2}\]
−(x − 3) + 4 < 5 − 2x
\[\frac{2\left( x - 1 \right)}{5} \leq \frac{3\left( 2 + x \right)}{7}\]
\[\frac{2x - 3}{3x - 7} > 0\]
\[\frac{1}{x - 1} \leq 2\]
\[\frac{5x + 8}{4 - x} < 2\]
\[\frac{x}{x - 5} > \frac{1}{2}\]
Solve each of the following system of equations in R.
2 (x − 6) < 3x − 7, 11 − 2x < 6 − x
Solve each of the following system of equations in R.
10 ≤ −5 (x − 2) < 20
Solve each of the following system of equations in R. \[\frac{4}{x + 1} \leq 3 \leq \frac{6}{x + 1}, x > 0\]
Solve
\[\left| x + \frac{1}{3} \right| > \frac{8}{3}\]
Solve \[\frac{1}{\left| x \right| - 3} < \frac{1}{2}\]
Solve \[\left| x - 1 \right| + \left| x - 2 \right| + \left| x - 3 \right| \geq 6\]
Solve \[1 \leq \left| x - 2 \right| \leq 3\]
Mark the correct alternative in each of the following:
If − 3x\[+\]17\[< -\]13, then
Solve the inequality, 3x – 5 < x + 7, when x is an integer.
Solve for x, `(|x + 3| + x)/(x + 2) > 1`.
If `|x - 2|/(x - 2) ≥ 0`, then ______.
If |x + 3| ≥ 10, then ______.
If a < b and c < 0, then `a/c` ______ `b/c`.
If |3x – 7| > 2, then x ______ `5/3` or x ______ 3.
Solve for x, the inequality given below.
`4/(x + 1) ≤ 3 ≤ 6/(x + 1)`, (x > 0)
The water acidity in a pool is considerd normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.
Given that x, y and b are real numbers and x < y, b < 0, then ______.
If –3x + 17 < –13, then ______.
If |x − 1| > 5, then ______.
If x > – 5, then 4x ______ –20.
