Solve the following inequalities.
\left| 5x-1 \right| \lt 3
Consider the following inequality:
\left| 5x-1\right| \lt 3
It is equivalent to:
-3 \lt 5x-1 \lt 3
Adding 1 to all sides gives:
-2 \lt 5x \lt 4
Dividing all sides by 5 gives:
-\dfrac{2}{5} \lt x \lt \dfrac{4}{5}
x is a solution if and only if -\dfrac{2}{5} \lt x \lt \dfrac{4}{5}.
\left| 2x+7 \right| \leqslant9
Consider the following inequality:
\left| 2x+7 \right| \leqslant9
It is equivalent to:
-9 \leqslant 2x+7 \leqslant 9
Subtracting 7 from all sides gives:
-16 \leqslant 2x \leqslant 2
Dividing all sides by 2 gives:
-8\leqslant x\leqslant1
x is a solution if and only if -8\leqslant x\leqslant1.
\left| x+3 \right| \gt 10
Consider the following inequality:
\left| x+3 \right| \gt 10
It is equivalent to:
x+3 \gt 10 or x+3 \lt -10
Subtracting 3 from all sides in the two equations gives:
x \gt 7 or x \lt -13
x is a solution if and only if x \lt -13 or x \gt 7.
\left| \dfrac{2x-3}{7}\right| \lt 5
Consider the following inequality:
\left| \dfrac{2x-3}{7}\right| \lt 5
It is equivalent to:
-5 \lt \dfrac{2x-3}{7} \lt 5
Multiplying all sides by 7 gives:
-35 \lt 2x-3 \lt 35
Adding 3 to all sides gives:
-32 \lt 2x \lt 38
Dividing all sides by 2 gives:
-16 \lt x \lt 19
x is a solution if and only if -16 \lt x \lt 19.
\left| \dfrac{x}{3}-4\right| \geqslant1
Consider the following inequality:
\left| \dfrac{x}{3}-4\right| \geqslant1
It is equivalent to:
\dfrac{x}{3}-4\geq 1 or \dfrac{x}{3} - 4 \leq -1
Adding 4 to all sides in the two equations gives:
\dfrac{x}{3}\geq5 or \dfrac{x}{3}\leq3
Multiplying all sides by 3 in the two equations gives:
x\geq15 or x\leq9
x is a solution if and only if x\leqslant9 or x\geqslant15.
\left| \dfrac{2x-5}{3}+6 \right| \lt 1
Consider the following inequality:
\left| \dfrac{2x-5}{3}+6 \right| \lt 1
It is equivalent to:
-1 \lt \dfrac{2x-5}{3}+6 \lt 1
Subtracting 6 from all sides gives:
-7 \lt \dfrac{2x-5}{3} \lt -5
Multiplying all sides by 3 gives:
-21 \lt 2x-5 \lt -15
Adding 5 to all sides gives:
-16 \lt 2x \lt -10
Dividing all sides by 2 gives:
-8 \lt x \lt -5
x is a solution if and only if -8 \lt x \lt -5.
\left| -x+5 \right| \lt 3
Consider the following inequality:
\left| -x+5 \right| \lt 3
It is equivalent to:
-3 \lt -x+5 \lt 3
Subtracting 5 from all sides gives:
-8 \lt -x \lt -2
Multiplying all sides by -1 gives:
8 \gt x \gt 2
Which is equivalent to:
2 \lt x \lt 8
x is a solution if and only if 2 \lt x \lt 8.