9000026406
Level:
B
Assuming \(x\in (-3;2)\),
write the following equation in the form which does not contain an absolute value.
\[
|x + 3| = |x - 2|
\]
\(x + 3 = -x + 2\)
\(x + 3 = x - 2\)
\(- x - 3 = -x + 2\)
\(- x - 3 = x + 2\)
Absolute value equations and inequalities
9000027307
Level:
A
Identify the solution set of the following inequality.
\[
|2x - 6|\leq 3
\]
\(\left [ 1.5;4.5\right ] \)
\(\left [ 0;6\right ] \)
\(\left (2;4\right )\)
\(\left [ -1;5\right ] \)
9000026407
Level:
B
Assuming \(x\in \left [ \frac{1}
{2};\infty \right )\),
write the following equation in the form which does not contain an absolute value.
\[
|x + 1| + |2x - 1| = 3
\]
\(x + 1 + 2x - 1 = 3\)
\(x + 1 - 2x - 1 = 3\)
\(x + 1 - 2x + 1 = 3\)
\(- x - 1 + 2x - 1 = 3\)
9000027309
Level:
A
Identify the solution set of the following inequality.
\[
|3x + 2| < -1
\]
\(\emptyset \)
\(\left (1;3\right )\)
\(\left [ -1;3\right ] \)
\(\left [ -2;0\right ] \)
9000026409
Level:
B
Consider the following equation.
\[
|2x - 4| = 5x - 7
\]
Solving the equation on the intervals where it is possible to evaluate the absolute
value we get equations on partial subintervals as follows.
\[\begin{aligned}
\text{for }x &\in (-\infty ;2)\colon &\text{for }x &\in [ 2;\infty )\colon & & & &
\\ - 2x + 4 & = 5x - 7 &2x - 4 & = 5x - 7 & & & &
\\ - 7x & = -11 & - 3x & = -3 & & & &
\\x & = \frac{11}
{7} &x & = 1 & & & &
\end{aligned}\]
Find the solution set of the original equation.
\(\left \{\frac{11}
{7} \right \}\)
\(\left \{\frac{11}
{7} ;1\right \}\)
\(\left \{1\right \}\)
\(\emptyset \)
9000027310
Level:
A
Identify the solution set of the following inequality.
\[
|2x + 11| > 0
\]
\(\left (-\infty ;-5.5\right )\cup \left (-5.5;\infty \right )\)
\(\left (-2;11\right )\)
\(\left (-\infty ;-11\right )\cup \left (11;\infty \right )\)
\(\emptyset \)
9000027308
Level:
A
Identify the solution set of the following inequality.
\[
|2x - 1| > 5
\]
\(\left (-\infty ;-2\right )\cup \left (3;\infty \right )\)
\(\left (-\infty ;-4.5\right )\cup \left (5.5;\infty \right )\)
\(\left (1.5;\infty \right )\)
\(\left (-\infty ;0\right )\cup \left [ 5;\infty \right )\)
9000027301
Level:
A
Identify the solution set of the following inequality.
\[
|x| < 3
\]
\(\left (-3;3\right )\)
\(\left (0;3\right )\)
\(\left [ -3;3\right ] \)
\(\left [ 0;3\right ] \)
9000027302
Level:
A
Identify the solution set of the following inequality.
\[
|x|\geq 5
\]
\(\left (-\infty ;-5\right ] \cup \left [ 5;\infty \right )\)
\(\left (-5;5\right )\)
\(\left (-\infty ;5\right )\cup \left (5;\infty \right )\)
\(\left [ -5;\infty \right )\)
9000027303
Level:
A
Identify the solution set of the following inequality.
\[
|x - 4|\leq 1
\]
\(\left [ 3;5\right ] \)
\(\left (-5;-3\right )\)
\(\left [ -5;-3\right ] \)
\(\left [ -1;4\right ] \)