Exercises 3

This page allows you to practice some exercises on Algebra. If you notice that you have difficulties, we advise you to go over the corresponding sections in the book.

These questions correspond to the following sections: - Section 2.7 - Intervals and Absolute Values - Section 2.8 - Sign Diagrams - Section 2.9 - Summation Notation - Section 2.10 - Rules for Sums - Section 2.11 - Newton's Binomial Formula - Section 2.12 - Double Sums

Question 18

Determine $x$ such that the following expressions hold true:

(a) $|3 − 2x| = 5$

(b) $|x| ≤ 2$

(c) $|x − 2| ≤ 1$

(d) $|3 − 8x| ≤ 5$

(e) $|x| > \sqrt{2}$

(f) $|x^2 − 2| ≤ 1$

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(a) $x = −1$ and $x = 4$.

(b) $−2 ≤ x ≤ 2$.

(c) $1 ≤ x ≤ 3$.

(d) $−1/4 ≤ x ≤ 1$.

(e) $x > \sqrt{2}$ or $x < − \sqrt{2}$.

(f) $−1 ≤ x^2 − 2 ≤ 1$, so $1 ≤ x^2 ≤ 3$, implying that $−\sqrt{3} ≤ x ≤ −1$ or $1 ≤ x ≤ \sqrt{3}$.

Question 19

A customer orders an iron bar whose advertised length is $5$ metres, but with a tolerance of $1$ mm. That is, the bar’s length may not deviate by more than $1$ mm from what is stipulated. Write a specification for the bar’s acceptable length $x$ in metres:

(a) by using a double inequality;

(b) with the aid of an absolute-value sign.

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(a) $4.999 < x < 5.001$.

(b) $|x - 5| < 0.001$.

Question 20

Solve the following inequalities:

(a) $2 < \frac{3x + 1}{2x + 4}$

(b) $\frac{120}{n} + 1.1 ≤ 1.85$

(c) $g^2 − 2g ≤ 0$

(d) $\frac{1}{p − 2} + \frac{3}{p^2 − 4p + 4} ≥ 0$

(e) $\frac{x+2}{x-1} < 0$

(f) $(x-1)(x+4) > 0$

(g) $(x-1)^2 (x+4) > 0$

(h) $(x-1)^3 (x-2) \leq 0$

(i) $\frac{x-3}{x+3} < 2x - 1$

(j) $x^3 + 2x^2 + x \leq 0$

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(a) $−7 < x < −2$.

(b) $n ≥ 160$ or $n < 0$.

(c) $0 ≤ g ≤ 2$.

(d) $p ≥ −1$ and $p \neq 2$.

(e) $−2 < x < 1$.

(f) $x < −4$ or $x > 1$.

(g) $x > −4$ and $x \neq 1$.

(h) $1 ≤ x ≤ 2$.

(i) $−3 < x < −2$ or $x > 0$.

(j) $x \leq 0$.

Question 21

Solve the inequality $\left( \frac{1}{x} − 1 \right) \div \left( \frac{1}{x} + 1 \right) ≥ 1$.

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$-1 < x < 0$.

Question 22

Evaluate the following sums:

(a) $\sum_{i=1}^{10} i$

(b) $\sum_{k=2}^{6} (5 \cdot 3^{k−2} − k)$

(c) $\sum_{m=0}^{5} (2m + 1)$

(d) $\sum_{l=0}^{2} 2^{2^l}$

For the following sums, expand them:

(e) $\sum_{k=−2}^{2} 2\sqrt{k + 2}$

(f) $\sum_{i=0}^{3} (x + 2i)^2$

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(a) $1 + 2 + 3 + \ldots + 10 = 55$.

(b) $(5 \cdot 3^0 − 2) + (5 \cdot 3^1 − 3) + (5 \cdot 3^2 − 4) + (5 \cdot 3^3 − 5) + (5 \cdot 3^4 − 6) = 585$.

(c) $1 + 3 + 5 + 7 + 9 + 11 = 36$.

(d) $2^{2^0} + 2^{2^1} + 2^{2^2} = 2^1 + 2^2 + 2^4 = 22$.

(e) $2\sqrt{0} + 2\sqrt{1} + 2\sqrt{2} + 2\sqrt{3} + 2\sqrt{4} = 2(3 + \sqrt{2} + \sqrt{3})$.

(f) $(x + 0)^2 + (x + 2)^2 + (x + 4)^2 + (x + 6)^2 = 4(x^2 + 6x + 14).$

Question 23

Express the following sums in summation notation:

(a) $4 + 8 + 12 + 16 + \cdots + 4n$

(b) $1^3 + 2^3 + 3^3 + 4^3 + \cdots+ n^3$

(c) $1 − \frac{1}{3} + \frac{1}{5} − \frac{1}{7} + \cdots+ (−1)^n \frac{1}{2n+1}$

(d) $a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj}$

(e) $a^3_i b_{i+3} + a^4_{i+1} b_{i+4} + \cdots + a^{p+3}_{i+p} b_{i+p+3}$

Show answer

(a) $\sum_{k=1}^{n} 4k$.

(b) $\sum_{k=1}^{n} k^3.$

(c) $\sum_{k=0}^{n} (-1)^{k} \frac{1}{2k + 1}$.

(d) $\sum_{k=1}^{n} a_{ik}b_{kj}$.

(e) $\sum_{k=0}^{p} a_{i+k}^{k+3}b_{i+k+3}$.

Question 24

Insert the appropriate limits of summation in the right-hand side of the following sums:

(a) $\sum_{k=1}^{10} (k − 2)t^k = \sum_{m=?}^{?} mt^{m+2}$

(b) $\sum_{n=0}^{N} 2^{n+5} = \sum_{j = ?}^{?} 32 \cdot 2^{j−1}$

Show answer

(a) $\sum_{k=1}^{10} (k − 2)t^k = \sum_{m=-1}^{8} mt^{m+2}$.

(b) $\sum_{n=0}^{N} 2^{n+5} = \sum_{j = 1}^{N+1} 32 \cdot 2^{j−1}$ (as $2^5 = 32$).

Question 25

Since early $2020$, the European Economic Area (EEA) consists of $30$ nations, who have agreed in principle to the free mobility of persons throughout the area. For the year $2025$, let $c_{ij}$ denote an estimate of the number of persons who will move from nation $i$ to nation $j$, for each $i \neq j$. If, say, $i = 25$ and $j = 10$, then we write $c_{25,10}$ for $c_{ij}$. Explain the meaning of the two sums:

(a) $\sum_{j=1}^{30} c_{ij} \quad \text{and} \quad$ (b) $\sum_{i=1}^{30} c_{ij}$.

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(a) The total number of people moving from nation $i$ within the EEA.

(b) The total number of people moving to nation $j$ within the EEA.

Question 26

Decide which of the following equalities are generally valid:

(a) $\sum_{k=1}^{n} ck^2 = c \sum_{k=1}^{n} k^2$

(b) $\left( \sum_{i=1}^{n} a_i \right)^2 = \sum_{i=1}^{n} a_i^2$

(c) $\sum_{j=1}^{n} b_j + \sum_{j=n+1}^{N} b_j = \sum_{j=1}^{N} b_j$

(d) $\sum_{k=3}^{7} 5^{k−2} = \sum_{k=0}^{4} 5^{k+1}$

(e) $\sum_{i=0}^{n-1} a_{i,j}^{2} = \sum_{k=1}^{n} a^2_{k−1,j}$

(f) $\sum_{k=1}^{n} a_k/k = \frac{1}{k} \sum_{k=1}^{n} a_k$

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Always true: (a), (c), (d), and (e).

Generally not true: (b) and (f).

Question 27

Expand and compute the following double sum: $\sum_{i=1}^{3} \sum_{j=1}^{4} i \cdot 3^{j}$.

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$\sum_{i=1}^{3} \sum_{j=1}^{4} i \cdot 3^{j} = \sum_{i=1}^{3} (i \cdot 3 + i \cdot 9 + i \cdot 27 + i \cdot 81) = \sum_{i=1}^{3} 120i = 720.$

Question 28

Consider a group of individuals each having a certain number of units of $m$ different goods. Let $a_{ij}$ denote the number of units of good $i$ owned by person $j$, for $i = 1, \ldots , m$ and for $j = 1, \ldots , n$. Explain in words the meaning of the following sums:

(a) $\sum_{j=1}^{n} a_{ij}$

(b) $\sum_{i=1}^{m} a_{ij}$

(c) $\sum_{j=1}^{n} \sum_{i=1}^{m} a_{ij}$

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(a) The total number of units of good $i.$

(b) The total number of units of all goods owned by person $j.$

(c) The total number of units of goods owned by the group as a whole.