Advertisements
Advertisements
प्रश्न
Solve \[\left| x + 1 \right| + \left| x \right| > 3\]
उत्तर
\[\text{ We have }, \left| x + 1 \right| + \left| x \right| > 3\]
\[\text{ As }, \left| x + 1 \right| = \binom{\left( x + 1 \right), x \geq - 1}{ - \left( x + 1 \right), x < - 1}\]
\[\text{ and } \left| x \right| = \binom{x, x \geq 0}{ - x, x < 0}\]
\[\text{ Case I: When } x < - 1, \]
\[\left| x + 1 \right| + \left| x \right| > 3\]
\[ \Rightarrow - \left( x + 1 \right) - x > 3\]
\[ \Rightarrow - 2x - 1 > 3\]
\[ \Rightarrow - 2x > 4\]
\[ \Rightarrow x < \frac{4}{- 2}\]
\[ \Rightarrow x < - 2\]
\[\text{ So }, x \in \left( - \infty , - 2 \right)\]
\[\text{ Case II: When } - 1 \leq x < 0, \]
\[\left| x + 1 \right| + \left| x \right| > 3\]
\[ \Rightarrow \left( x + 1 \right) - x > 3\]
\[ \Rightarrow 1 > 3, \text{ which is not possible }\]
\[\text{ So }, x \in \phi\]
\[\text{ Case III: When x } \geq 0, \]
\[\left| x + 1 \right| + \left| x \right| > 3\]
\[ \Rightarrow \left( x + 1 \right) + x > 3\]
\[ \Rightarrow 2x + 1 > 3\]
\[ \Rightarrow 2x > 2\]
\[ \Rightarrow x > \frac{2}{2}\]
\[ \Rightarrow x > 1\]
\[\text{ So }, x \in \left( 1, \infty \right)\]
\[\text{ From all the cases, we get }\]
\[x \in \left( - \infty , - 2 \right) \cup \left( 1, \infty \right)\]
APPEARS IN
संबंधित प्रश्न
Solve: −4x > 30, when x ∈ R
x + 5 > 4x − 10
3x + 9 ≥ −x + 19
\[\frac{4 + 2x}{3} \geq \frac{x}{2} - 3\]
\[\frac{2x + 3}{5} - 2 < \frac{3\left( x - 2 \right)}{5}\]
\[\frac{x}{x - 5} > \frac{1}{2}\]
Solve each of the following system of equations in R.
1. x + 3 > 0, 2x < 14
2x + 6 ≥ 0, 4x − 7 < 0
Solve each of the following system of equations in R.
2x − 3 < 7, 2x > −4
Solve each of the following system of equations in R.
2x + 5 ≤ 0, x − 3 ≤ 0
Solve each of the following system of equations in R.
3x − 1 ≥ 5, x + 2 > −1
Solve each of the following system of equations in R.
\[\frac{2x - 3}{4} - 2 \geq \frac{4x}{3} - 6, 2\left( 2x + 3 \right) < 6\left( x - 2 \right) + 10\]
Solve each of the following system of equations in R.
\[\frac{7x - 1}{2} < - 3, \frac{3x + 8}{5} + 11 < 0\]
Solve each of the following system of equations in R.
10 ≤ −5 (x − 2) < 20
Solve
\[\left| x + \frac{1}{3} \right| > \frac{8}{3}\]
Solve
\[\left| 4 - x \right| + 1 < 3\]
Solve \[\frac{\left| x + 2 \right| - x}{x} < 2\]
Solve \[1 \leq \left| x - 2 \right| \leq 3\]
Mark the correct alternative in each of the following:
If − 3x\[+\]17\[< -\]13, then
Mark the correct alternative in each of the following:
If x is a real number and \[\left| x \right|\]\[<\]5, then
Mark the correct alternative in each of the following:
\[\left| x - 1 \right|\]\[>\]5, then
Mark the correct alternative in each of the following:
The linear inequality representing the solution set given in 
Mark the correct alternative in each of the following:
The solution set of the inequation \[\left| x + 2 \right|\]\[\leq\]5 is
Solve the inequality, 3x – 5 < x + 7, when x is a natural number.
Solve for x, `(|x + 3| + x)/(x + 2) > 1`.
Solve the following system of inequalities:
`x/(2x + 1) ≥ 1/4, (6x)/(4x - 1) < 1/2`
The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then ______.
If –x ≤ –4, then 2x ______ 8.
If |3x – 7| > 2, then x ______ `5/3` or x ______ 3.
Solve for x, the inequality given below.
`-5 ≤ (2 - 3x)/4 ≤ 9`
If x is a real number and |x| < 3, then ______.
If |x + 2| ≤ 9, then ______.
If `(-3)/4 x ≤ – 3`, then x ______ 4.
If x > – 5, then 4x ______ –20.
If – 2x + 1 ≥ 9, then x ______ – 4.
