# JIT 4. Operations with Real Numbers (1.1)

## **Adding Two negative numbers:**

$$-5 - 26 = ?$$

First add the associated positives ('absolute values'):

$$5 + 26 = 31$$

Final answer has a negative:

$$-5 - 26 = -31$$

- **Examples.**

$$-14 - 17 =-31$$

$$-50 - 74 =-124$$

$$-48 - 57 =-105$$

## **Adding Negative and positive numbers:**

$$9 - 36 = ?$$

Subtract in opposite order:

$$36 - 9 = 27$$

In original order it has a minus sign:

$$9 - 36 = -27$$

- **Examples.**

$$3 - 8 =-5$$

$$14 - 20 =-6$$

$$11 - 17 =-6$$

## **Subtracting negatives.**

$$4 - (-8) = ?$$

A double negative is positive; then add:

$$4 - (-8) = 4 + 8 = 12$$

- **Examples.**

$$7 - (-15) =7+15=22$$

$$-4 - (-8) =-4+8=4$$

$$-7 - (-1) =-7+1=-6$$

## **Multiplication.**

$$
(-3)(-5) = ?
$$

$$
(-3)(5) = ?
$$

$$
(3)(-5) = ?
$$

- **Solution.**
Multiply 3 and 5 first:

$$
(3)(5) = 15
$$

Opposite signs give a negative:

$$
(-3)(5) = -15
$$

$$
(3)(-5) = -15
$$

Same signs give a positive:

$$
(3)(5) = 15
$$

$$
(-3)(-5) = 15
$$

- **Examples.**

$$
(-7)(3) =-21
$$

$$
(5)(-6) =-30
$$

$$
(-2)(-9) =18
$$

## **Division.**

$$
\frac{8}{4} = ?
$$

$$
\frac{-8}{4} = ?
$$

$$
\frac{8}{-4} = ?
$$

$$
\frac{-8}{-4} = ?
$$

- **Solution.** Divide 8 and 4 first:

$$
\frac{8}{4} = 2
$$

Opposite signs give a negative:

$$
\frac{8}{-4} = -2
$$

$$
\frac{-8}{4} = -2
$$

Same signs give a positive:

$$
\frac{-8}{-4} = 2
$$

Notice

$$
\frac{8}{-4} = \frac{-8}{4} = \frac{8}{4} = -2
$$

and in general:

$$
\frac{x}{-y} = \frac{-x}{y} = -\frac{x}{y}
$$

- **Examples.**

$$\dfrac{15}{-5}=-3$$

$$\dfrac{-20}{-4}=5$$

$$\dfrac{-18}{3}=-6$$

## **JIT 19. Fractions: multiplication.**

$$
\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}
$$

- **Examples.**

$$
\frac{2}{7} \cdot \frac{4}{3} =\dfrac{2 \cdot 4}{7 \cdot 3}=\dfrac{8}{21}
$$

$$
\left( -\frac{2}{3} \right) \cdot \left( \frac{4}{7} \right) =\dfrac{(-2) \cdot (4)}{(3)(7)}=-\dfrac{8}{21}
$$

$$
\left( -\frac{3}{4} \right) \cdot \left( -\frac{3}{7} \right) =\dfrac{(-3)(-3)}{(4)(7)}=\dfrac{9}{28}
$$

## **Fractions: division.**

$$
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
$$

- **‘Flip and multiply.’**

- **Examples.**

$$
\frac{2}{7} \div \frac{4}{3} =\frac{2}{7} \cdot \dfrac{3}{4}=\dfrac{6}{28}=\dfrac{3}{14}
$$

$$
\dfrac{-\frac{2}{3} }{ \frac{5}{7} }=\left(-\dfrac{2}{3}\right) \div \dfrac{5}{7} = \left(-\dfrac{2}{3}\right) \cdot \dfrac{7}{5} = \dfrac{-14}{15}
$$

$$\dfrac{-\frac{3}{4}}{-\frac{2}{7}}=\left(-\dfrac{3}{4} \right) \div \left(-\dfrac{2}{7} \right)=\left(-\dfrac{3}{4} \right) \cdot \left(-\dfrac{7}{2} \right)=\dfrac{21}{8}$$

## **Fractions: special cases.**

$$
7 \cdot \left(\frac{4}{3}\right) = \frac{7}{1} \cdot \left(\frac{4}{3}\right) =
$$

$$
5 \cdot \left(\frac{7}{2}\right) =
$$

$$
3 \cdot \left(-\frac{2}{5}\right) =
$$

$$
\frac{7}{4} \div \frac{3}{3} = \frac{7}{1} \div \frac{4}{3} =
$$

$$
\frac{3}{-5} \div \frac{2}{2} =
$$

$$
\left(\frac{7}{3}\right) \div 4 = \frac{7}{3} \div \frac{4}{1} =
$$

$$
\left(\frac{2}{3}\right) \div -5 =
$$

- **Fractions: addition and subtraction.**
- **Get common denominators.**
- **Key fact:**

$$
\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}
$$

- **Key fact:**

$$
\frac{a}{b} = \frac{a \cdot c}{b \cdot c}
$$

- **Example.**

$$
\frac{2}{3} - \frac{4}{5} = ?
$$

- **Solution.** Write

$$
\frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}
$$

and

$$
\frac{4}{5} = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15}
$$

So

$$
\frac{2}{3} - \frac{4}{5} = \frac{10}{15} - \frac{12}{15} = \frac{10 - 12}{15} = \frac{-2}{15} = -\frac{2}{15}
$$

- **We say 15 is the ‘common denominator’ of the two fractions.**

- **Examples.**

$$
\frac{3}{4} - \frac{4}{5} =
$$

$$
\frac{-2}{5} - \frac{1}{2} =
$$

$$
\frac{-1}{3} + \frac{2}{7} =
$$

$$
7 - \frac{4}{5} =
$$

## **Reduction of fractions.**
- **Recall key fact:**

$$
\frac{a}{b} = \frac{a \cdot c}{b \cdot c}
$$

$$
\frac{15}{10} = \frac{3 \cdot 5}{2 \cdot 5} = \frac{3}{2}
$$

$$
\frac{24}{16} =
$$

$$
\frac{12}{18} =
$$

$$
\frac{44}{46} =
$$

$$
\frac{45}{65} =
$$

## **Multiplication and division with decimals.**
- **Multiplication by a power of 10 shifts the decimal point.**

- **Examples.**

$$
(10)(2.65) = 26.5
$$

$$
(100)(.0012) = .12
$$

$$
(1000)(-0.01845) = -18.45
$$

- Multiplication by 10 shifts the decimal point one place to the right.
- Multiplication by 100 shifts the decimal point two places to the right.
- Multiplication by 1000 shifts the decimal point three places to the right.
- Etc.

- **Examples.**

$$
(10)(1.87) =
$$

$$
(100)(-0.245) =
$$

$$
(1000)(1.673) =
$$

$$
(10000)(0.03654) =
$$

## **Division with decimals.**

$$
2.65 \div 10 = \frac{2.65}{10} = 0.265
$$

$$
0.12 \div 100 = \frac{0.12}{100} = 0.0012
$$

$$
18.54 \div 1000 = \frac{18.54}{1000} = 0.01854
$$

- Division by 10 shifts the decimal point one place to the left.
- Division by 100 shifts the decimal point two places to the left.
- Division by 1000 shifts the decimal point three places to the left.
- Etc.

- **Examples**

$$
87.16 \div 10 =
$$

$$
0.0463 \div 100 =
$$

$$
-4.623 \div 1000 =
$$

## **JIT 9. Order of operations: PEMDAS.** (With Section 1.1)
1. Do everything in **P**arentheses first
2. Do **E**xponents next
3. Do **MD** (multiplication/division) next, left to right
4. Do **AS** (addition/subtraction) next, left to right

- **Examples.**

$$
4 \cdot 3 + 4^2 \cdot 2 - 2(8 - 5) =
$$

$$
4(2 + 3 \cdot 2^3 + 6 - 3 - 2) =
$$

$$
3 \cdot (4 - 2)^2 - 2(5 + 4) - 7 =
$$

## **JIT 14. Equation Solving Principles.** (For use with section 1.1, section 1.3, and section 1.5)
- **Principle:** Isolate the variable $x$.

- **Examples.**

### A. Solve

$$
7x + 9 = 26
$$

- **Solution.** Subtract 9 from both sides of the equation:

$$
7x = 26 - 9
$$

$$
7x = 17
$$

Divide both sides by 7:

$$
x = \frac{17}{7}
$$

---

### B. Solve

$$
9y + 6 = 22y - 5
$$

- **Solution.** Gather like terms by subtracting $22y$ and subtracting 6:

$$
9y - 22y = -5 - 6
$$

$$
-13y = -11
$$

Divide both sides by $-13$:

$$
y = \frac{-13}{-11} = \frac{13}{11}
$$

---

### C. Solve

$$
3(2x - 5) + 14 = 2(x - 9)
$$

- **Solution.** Distribute:

$$
6x - 15 + 14 = 2x - 18
$$

Gather like terms:

$$
6x - 1 = 2x - 18
$$

Subtract $2x$, add 1:

$$
6x - 2x = -18 + 1
$$

$$
4x = -17
$$

Divide by 4:

$$
x = \frac{-17}{4}
$$

- **Methods of solution.**
- Distribute
- Gather like terms
- Add/subtract the same number to both sides
- Multiply/divide the same number to both sides

### D. Solve

$$
2y - 7 = -18
$$

**Solution:**
\begin{align}
2y - 7 &= -18 \\
2y-7+7&=-18+7 \\
2y&=-11 \\
\dfrac{2y}{2}&=\dfrac{-11}{2} \\
y&=-\dfrac{11}{2}
\end{align}

---

### E. Solve

$$
2x + 5 = -7x + 20
$$

**Solution:**
\begin{align}
2x + 5 &= -7x + 20 \\
2x+7x + 5 &= -7x+7x + 20 \\
9x+5&=20 \\
9x+5-5&=20-5 \\
9x&=15 \\
\dfrac{9x}{9}&=\dfrac{15}{9} \\
x&=\dfrac{5 \cdot 3}{3 \cdot 3} \\
x&=\dfrac{5}{3}.
\end{align}

---

### F. Solve

$$
4(x - 2) - 13 = 3(3x + 7)
$$

**Solution:**
\begin{align}
4(x - 2) - 13 &= 3(3x + 7) \\
4 \cdot x +4 \cdot (-2) - 13 &= 3 \cdot (3x)+3 \cdot (7) \\
4x-8 - 13 &= 9x+21 \\
4x-21 &= 9x+21 \\
4x-9x-21 &= 9x-9x+21 \\
-5x-21 &= 21 \\
-5x-21+21 &= 21+21 \\
-5x&=42 \\
\dfrac{-5x}{-5}&=\dfrac{42}{-5} \\
x&=-\dfrac{42}{5}
\end{align}