This worksheet has 30 problems grouped into the three performance levels from the national assessment standards (basic – consolidated – advanced). Each section has calculations, properties, equations and inequalities. Write complete solutions; the criteria used to grade a solution are given below.
Evaluation criteria
| Level | Items | Performance descriptors – the student: |
|---|---|---|
| Basic | 1 – 10 | computes the absolute value, the integer part and the fractional part of given numbers (including negative or irrational ones); recognizes the basic properties; solves equations and inequalities of the form $|ax + b| = c$, $|ax + b| \leq c$, $\lbrack x\rbrack = k$, in a familiar mathematical context. |
| Consolidated | 11 – 22 | removes the absolute value after determining the sign; justifies properties; solves equations of the type $|E| = |F|$, $|E| = F$ (with a condition), sums of absolute values (the interval method), inequalities whose solutions are unions of intervals, and equations/inequalities with $\left\lbrack E(x) \right\rbrack$, in varied contexts. |
| Advanced | 23 – 30 | proves inequalities using the properties of absolute value; discusses the number of solutions depending on a parameter; solves equations with nested absolute values, equations of the type $\left\lbrack E(x) \right\rbrack = F(x)$ or with $\lbrack x\rbrack$ and $\left\{ x \right\}$; generalizes a calculation, in a complex context. |
A complete solution meets: C1 – identifying the type of problem and writing the conditions (the sign of the expression, the right-hand side non-negative, $k \in {\mathbb{Z}}$, $\left\{ x \right\} \in \lbrack 0,\ 1)$); C2 – correctly applying the definitions and properties; C3 – the correctness of the calculations; C4 – the write-up: intersecting with the conditions, writing the solution set $S$ (a finite set, an interval or a union of intervals) and checking it.
Section A – Basic level (items 1 – 10)
Write complete solutions.
Calculations
-
Calculate $| - 7| + |3 - 8| - | - 2| \cdot |4|$.
-
Show that $\left| \sqrt{5} - 3 \right| + \left| \sqrt{5} - 2 \right| = 1$.
-
Calculate $\left\lbrack \text{3,7} \right\rbrack$, $\left\lbrack - \text{3,7} \right\rbrack$, $\left\{ \text{3,7} \right\}$ and $\left\{ - \text{3,7} \right\}$.
-
Find $\left\lbrack \sqrt{20} \right\rbrack$, $\left\lbrack - \sqrt{20} \right\rbrack$ and $\left\{ \sqrt{20} \right\}$.
Properties
-
Determine whether each statement is true or false. Justify with a property or with a counterexample.
a) $|a + b| = |a| + |b|$, for any $a,\ b \in {\mathbb{R}}$;
b) $|x|^{2} = x^{2}$, for any $x \in {\mathbb{R}}$;
c) $\sqrt{x^{2}} = x$, for any $x \in {\mathbb{R}}$;
d) $\lbrack x + 2\rbrack = \lbrack x\rbrack + 2$, for any $x \in {\mathbb{R}}$;
e) $\lbrack 2x\rbrack = 2\lbrack x\rbrack$, for any $x \in {\mathbb{R}}$.
-
Show that $|x - 3| + |x + 1| = 4$, for any $x \in \lbrack - 1,\ 3\rbrack$.
Equations
-
Solve in $\mathbb{R}$ the equation $|x - 2| = 5$.
-
Solve in $\mathbb{R}$ the equations:
a) $|2x + 1| = - 3$;
b) $|3x - 6| = 0$.
-
Find the real numbers $x$ for which $\lbrack x\rbrack = - 2$.
Inequalities
-
Solve in $\mathbb{R}$ the inequalities and write the solution set as an interval:
a) $|x - 1| \leq 3$;
b) $\lbrack x\rbrack < 3$.
Section B – Consolidated level (items 11 – 22)
Write complete solutions.
Calculations
-
Show that the number $N = \left| 1 - \sqrt{2} \right| + \left| \sqrt{2} - \sqrt{3} \right| + \left| \sqrt{3} - 2 \right|$ is a natural number.
-
Show that $\sqrt{x^{2} - 2x + 1} + \sqrt{x^{2} - 8x + 16} = 3$, for any $x \in \lbrack 1,\ 4\rbrack$.
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Calculate $S = \left\lbrack \sqrt{1} \right\rbrack + \left\lbrack \sqrt{2} \right\rbrack + \left\lbrack \sqrt{3} \right\rbrack + \ldots + \left\lbrack \sqrt{10} \right\rbrack$.
Properties
-
Let $a,\ b \in {\mathbb{R}}$ with $|a| \leq 1$ and $|b| \leq 1$. Show that:
a) $|a + b| \leq 2$;
b) $|ab| \leq 1$.
-
a) Show that $\lbrack x\rbrack + \lbrack - x\rbrack = - 1$, for any $x \in {\mathbb{R}} \smallsetminus {\mathbb{Z}}$.
b) Calculate $\left\lbrack \sqrt{3} \right\rbrack + \left\lbrack - \sqrt{3} \right\rbrack + \left\{ \sqrt{3} \right\} + \left\{ - \sqrt{3} \right\}$.
Equations
-
Solve in $\mathbb{R}$ the equation $|2x - 3| = |x + 1|$.
-
Solve in $\mathbb{R}$ the equation $|x - 3| = 2x - 1$.
-
Solve in $\mathbb{R}$ the equation $|x - 1| + |x - 2| = 3$.
-
Solve in $\mathbb{R}$ the equation $\left\lbrack \frac{2x + 1}{3} \right\rbrack = 2$.
Inequalities
-
Solve in $\mathbb{R}$ the inequality $|2x - 5| > 3$.
-
Consider the set $A = \{ x \in {\mathbb{R}}\text{ | }1 \leq |x - 2| < 4\}$.
a) Write the set $A$ as a union of intervals.
b) Find the set $A \cap {\mathbb{Z}}$.
-
Solve in $\mathbb{R}$ the inequality $\left| 2\lbrack x\rbrack - 1 \right| \leq 3$.
Section C – Advanced level (items 23 – 30)
Write complete solutions, with all justifications.
Properties and proofs
-
The real numbers $x$ and $y$ satisfy $|x - 2| \leq 1$ and $|y + 1| \leq 2$. Show that $|x + 2y| \leq 5$.
-
Consider the expression $E(x) = |x - 1| + |x + 2|$, where $x \in {\mathbb{R}}$.
a) Show that $E(x) \geq 3$, for any $x \in {\mathbb{R}}$, and find the values of $x$ for which $E(x) = 3$.
b) Discuss, depending on the values of the real parameter $m$, the number of solutions of the equation $E(x) = m$.
Equations
-
Solve in $\mathbb{R}$ the equation $\left| |x - 2| - 3 \right| = 1$.
-
Solve in $\mathbb{R}$ the equation $\lbrack x\rbrack = \frac{3x - 1}{2}$.
-
Solve in $\mathbb{R}$ the equation $\lbrack x\rbrack + 2\left\{ x \right\} = 3$.
Inequalities
-
Solve in $\mathbb{R}$ the inequality $|x + 1| - |x - 2| \geq 1$.
Synthesis problems
-
a) Check the equality $\lbrack x\rbrack + \left\lbrack x + \frac{1}{2} \right\rbrack = \lbrack 2x\rbrack$ for $x = \text{2,3}$ and for $x = \text{2,7}$, then prove that it holds for any $x \in {\mathbb{R}}$.
b) Solve in $\mathbb{R}$ the equation $\lbrack x\rbrack + \left\lbrack x + \frac{1}{2} \right\rbrack = 5$.
-
(Top-grade problem)
a) Show that, for any $k \in {\mathbb{N}}^{*}$, there are exactly $2k + 1$ natural numbers $n$ with $\left\lbrack \sqrt{n} \right\rbrack = k$.
b) Calculate $S = \left\lbrack \sqrt{1} \right\rbrack + \left\lbrack \sqrt{2} \right\rbrack + \left\lbrack \sqrt{3} \right\rbrack + \ldots + \left\lbrack \sqrt{99} \right\rbrack$.
Self-assessment
| Section | Items solved correctly | The criterion I missed most often (C1–C4) | What I should review from the summary |
|---|---|---|---|
| A – basic | …… out of 10 | ||
| B – consolidated | …… out of 12 | ||
| C – advanced | …… out of 8 |