I. Evaluating absolute values. Removing the absolute value bars
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Calculate:
a) $| - 7| + |3 - 8| - | - 2| \cdot | - 4|$;
b) $\left| 2 - \sqrt{5} \right| + \left| \sqrt{5} - 3 \right|$;
c) $\left| 1 - \sqrt{2} \right| + \left| \sqrt{2} - \sqrt{3} \right| + \left| \sqrt{3} - 2 \right|$.
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Show that the number $a = \left| 2\sqrt{3} - 4 \right| + \left| 2\sqrt{3} - 3 \right|$ is a natural number.
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Show that $|x - 2| + |x + 3| = 5$, for any $x \in \lbrack - 3,\ 2\rbrack$.
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Rewrite without the absolute value the expression $E(x) = |x - 3| + |2x + 4|$, where $x$ is a real number.
II. Properties of the absolute value
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Show that the number $N = \sqrt{\left( 1 - \sqrt{2} \right)^{2}} + \sqrt{\left( 3 - \sqrt{2} \right)^{2}}$ is a natural number.
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Find the real numbers $x$ for which:
a) $|x - 5| = x - 5$;
b) $|2x + 6| = - 2x - 6$.
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Find the real numbers $x$ and $y$ for which $|x - 2y| + |x + y - 6| = 0$.
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Show that $\sqrt{x^{2} - 4x + 4} + \sqrt{x^{2} - 10x + 25} = 3$, for any $x \in \lbrack 2,\ 5\rbrack$.
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Find the sets:
a) $A = \left\{ x \in {\mathbb{Z}} \mid |2x - 1| \leq 5 \right\}$;
b) $B = \left\{ x \in {\mathbb{R}} \mid |x + 2| > 3 \right\}$.
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Let $x$ be a real number such that $|x - 1| \leq 2$. Show that $|3x + 1| \leq 10$.
III. Equations with absolute values
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Solve in $\mathbb{R}$ the equations:
a) $|x - 3| = 5$;
b) $|2x + 1| = 7$;
c) $|3x - 2| = - 1$;
d) $\sqrt{x^{2} - 6x + 9} = 2$.
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Solve in $\mathbb{R}$ the equations:
a) $|x - 1| = |2x + 3|$;
b) $|x - 2| = 2x - 7$.
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Solve in $\mathbb{R}$ the equations:
a) $|x - 1| + |x - 3| = 4$;
b) $\left| |x - 2| - 3 \right| = 1$.
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Solve in $\mathbb{R}$ the equations:
a) $x^{2} - 5|x| + 6 = 0$;
b) $\left| x^{2} - 4 \right| = 3x$.
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Consider the expression $E(x) = |x - 1| + |x - 5|$, where $x$ is a real number.
a) Show that $E(x) \geq 4$, for any real number $x$.
b) Solve in $\mathbb{R}$ the equation $E(x) = 4$.