Solve for x the inequality \( \dfrac{4}{x+1} \le 3 \le \dfrac{6}{x+1} , (x>0) \)
\( \dfrac{1}{3} \le x \le 1 \)
Solve for x the inequality \( \dfrac{|x-2|-1}{|x-2|-2} \le 0 \)
[0,1] ∪ [3,4]
Solve for x the inequality \( \dfrac{1}{|x|-3} \le \dfrac{1}{2} \)
(−∞,−5) ∪ (−3,3) ∪ [5,∞)
Solve for x the inequalities \( |x−1| \le 5 , |x| \ge 2 \)
[−4,−2] ∪ [2,6]
Solve for x the inequality \( −5 \le \dfrac{2−3x}{4} \le 9 \)
[−\dfrac{34}{3}, \dfrac{22}{3}]
Solve for x the inequalities \( 4x+3 \ge 2x+17 , 3x−5 < −2 \)
No Solution
A company manufactures cassettes. Its cost and revenue functions are \( C(x)=26000+30x \) and \( R(x)=43x \). How many cassettes must be sold for the company to realise some profit?
More than 2000.
The water acidity in a pool is considered normal when the average pH of three readings is between 8.2 and 8.5. If first two readings are 8.48 and 8.35, find the range of the third reading.
Between 7.77 and 8.77.
A solution of 9% acid is to be diluted by adding 3% acid solution to it. If mixture is to be more than 5% but less than 7% acid and there are 460 litres of 9% solution, how many litres of 3% solution are needed?
More than 230 litres but less than 920 litres.
A solution is to be kept between 40°C and 45°C. What is the range in °F? Conversion: \( F = \dfrac{9}{5}C + 32 \)
Between 104°F and 113°F
The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest. If perimeter is more than 166 cm, find minimum length of shortest side.
41 cm.
The temperature at depth x km is given by \( T = 30 + 25(x−3) \). For \( 3 \le x \le 15 \), find depth at which temperature is between 155°C and 205°C.
Between 8 km and 10 km
Solve the following system of inequalities \(\dfrac{2x+1}{7x-1} > 5\) and \(\dfrac{x+7}{x-8} > 2\).
No Solution
Find the linear inequalities for which the shaded region in Fig 6.5 is the solution set. (The figure shows a triangular/ trapezoidal shaded region bounded by the lines \(x+y=20\), \(3x+2y=48\), the x-axis and the y-axis.)
\(x + y \le 20,\)
\(3x + 2y \le 48,\)
\(x \ge 0,\; y \ge 0\)
Find the linear inequalities for which the shaded region in Fig 6.6 is the solution set. (The figure shows a rectangular/ trapezoidal shaded region bounded by the lines \(x+y=8\), \(x+y=4\), \(x=5\), \(y=5\), and the coordinate axes.)
\(x + y \le 8,\)
\(x + y \ge 4,\)
\(x \le 5,\; y \le 5,\; x \ge 0,\; y \ge 0\)
Show that the following system of linear inequalities has no solution: \(x + 2y \le 3,\; 3x + 4y \ge 12,\; x \ge 0,\; y \ge 1\).
Solve the following system of linear inequalities: \(3x + 2y \ge 24,\; 3x + y \le 15,\; x \ge 4\).
No Solution
Show that the solution set of the following system of linear inequalities is an unbounded region: \(2x + y \ge 8,\; x + 2y \ge 10,\; x \ge 0,\; y \ge 0\).
If \(x < 5\), then
\(-x < -5\)
\(-x \le -5\)
\(-x > -5\)
\(-x \ge -5\)
Given that \(x, y\) and \(b\) are real numbers and \(x < y,\; b < 0\), then
\(\dfrac{x}{b} < \dfrac{y}{b}\)
\(\dfrac{x}{b} \le \dfrac{y}{b}\)
\(\dfrac{x}{b} > \dfrac{y}{b}\)
\(\dfrac{x}{b} \ge \dfrac{y}{b}\)
If \(-3x + 17 < -13\), then
\(x \in (10,\infty)\)
\(x \in [10,\infty)\)
\(x \in (-\infty,10]\)
\(x \in [-10,10]\)
If \(x\) is a real number and \(|x| < 3\), then
\(x \ge 3\)
\(-3 < x < 3\)
\(x \le -3\)
\(-3 \le x \le 3\)
\(x\) and \(b\) are real numbers. If \(b > 0\) and \(|x| > b\), then
\(x \in (-b,\infty)\)
\(x \in (-\infty,b]\)
\(x \in (-b,b)\)
\(x \in (-\infty,-b) \cup (b,\infty)\)
If \(|x-1| > 5\), then
\(x \in (-4,6)\)
\(x \in [-4,6]\)
\(x \in (-\infty,-4) \cup (6,\infty)\)
\(x \in (-\infty,-4) \cup [6,\infty)\)
If \(|x+2| \le 9\), then
\(x \in (-7,11)\)
\(x \in [-11,7]\)
\(x \in (-\infty,-7) \cup (11,\infty)\)
\(x \in (-\infty,-7) \cup [11,\infty)\)
The inequality representing the shaded square in Fig 6.7 (square centred at origin extending to \(\pm5\) on both axes) is:
\(|x| < 5\)
\(|x| \le 5\)
\(|x| > 5\)
\(|x| \ge 5\)
Which set corresponds to the number line in Fig 6.8?
\(x \in (-\infty,5)\)
\(x \in (-\infty,5]\)
\(x \in [5,\infty)\)
\(x \in (5,\infty)\)
Which set corresponds to the number line in Fig 6.9 (point at \(\tfrac{9}{2}\))?
\(x \in (\tfrac{9}{2},\infty)\)
\(x \in [\tfrac{9}{2},\infty)\)
\(x \in (-\infty,\tfrac{9}{2})\)
\(x \in (-\infty,\tfrac{9}{2}]\)
Which set corresponds to the number line in Fig 6.10 (open circle at \(\tfrac{7}{2}\) with arrow right)?
\(x \in (-\infty,\tfrac{7}{2})\)
\(x \in (-\infty,\tfrac{7}{2}]\)
\(x \in [\tfrac{7}{2},\infty)\)
\(x \in (\tfrac{7}{2},\infty)\)
Which set corresponds to the number line in Fig 6.11 (filled dot at \(-2\) with arrows both ways)?
\(x \in (-\infty,-2)\)
\(x \in (-\infty,-2]\)
\(x \in (-2,\infty)\)
\(x \in [-2,\infty)\)
If x < y and b < 0, then \( \dfrac{x}{b} < \dfrac{y}{b} \).
False
If xy > 0, then x > 0 and y < 0.
False
If xy > 0, then x < 0 and y < 0.
True
If xy < 0, then x < 0 and y < 0.
False
If x < -5 and x < -2, then x ∈ (−∞, −5).
True
If x < −5 and x > 2, then x ∈ (−5, 2).
False
If x > −2 and x < 9, then x ∈ (−2, 9).
True
If |x| > 5, then x ∈ (−∞, −5) ∪ [5, ∞).
False
If |x| ≤ 4, then x ∈ [−4, 4].
True
Graph of x < 3 is the shaded region to the left of the vertical line x = 3.
False
Graph of x ≥ 0 is the shaded region to the right of the y-axis.
True
Graph of y ≤ 0 is the shaded region above the x-axis.
False
Solution set of x ≥ 0 and y ≤ 0 is the shaded region in the second quadrant.
False
Solution set of x ≥ 0 and y ≤ 1 is the shaded region to the right of the y-axis and below the line y = 1.
True
Solution set of x + y ≥ 0 is the shaded region above the line x + y = 0.
True
If −4x ≥ 12, then x ____ −3.
≤
If −3/4 ≤ x ≤ −3, then x ____ 4.
≥
If 2/(x+2) > 0, then x ____ −2.
>
If x > −5, then 4x ____ −20.
>
If x > y and z < 0, then −xz ____ −yz.
>
If p > 0 and q < 0, then p − q ____ p.
>
If |x + 2| ≥ 5, then x ____ −7 or x ____ 3.
< , >
If −2x + 1 ≥ 9, then x ____ −4.
≤