Absolute value in equations #2

Short Quiz in Math 7 solves linear equation or inequality in one variable involving absolute value

CONCEPT OF INTEGERS What are INTEGERS? Integers are whole numbers that describe opposite ideas in mathematics. Integers can either be negative (-), positive (+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the value of the integer. POSITIVE INTEGERS Are numbers to the right of zero. Are valued greater than zero. Express ideas of up, a gain or a profit. The sign for a positive integer is (+), however the sign is not always needed. Meaning +3 is the same value as 3. NEGATIVE INTEGERS Are numbers to the left of zero. Are valued less than zero. Express ideas of down or a loss. The sign for a negative integer is (-). This sign is always needed. Opposite Numbers/Integers – are the pairs of integers that have the same absolute value or have the same distance away from zero. ABSOLUTE VALUE The distance of a number from the origin (0) regardless of direction is called absolute value. The absolute value of a number is never negative. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. Examples: I 4 I = 4, +4 is read “ the absolute value of 4 is 4 “ I -3 I = 3, -3 is read “ the absolute value of -3 is 3” - I 3 I = -3, means “ the negative of the absolute value of 3 is -3 “ COMPARING AND ARRANGING INTEGERS Integers can be compared using a number line. As you move to the left along the number line, the integers decrease in value. On the other hand, integers increase in value as you move to the right along the number line. To arrange integers in ascending order is to arrange them from least to greatest. This means that when you use the number line, the smallest the integer is to the left of 0 on the number line. To arrange integers in descending order is to arrange them from greatest to least. This means that when you use the number line, the largest the integer is to the right of 0 on the number line. This is read as “nine is greater than negative 12.” This is read as “negative thirteen is less than negative 5.” This is read as “negative eight is greater than negative 18.”

What are INTEGERS? Integers are whole numbers that describe opposite ideas in mathematics. Integers can either be negative (-), positive (+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the value of the integer. POSITIVE INTEGERS Are numbers to the right of zero. Are valued greater than zero. Express ideas of up, a gain or a profit. The sign for a positive integer is (+), however the sign is not always needed. Meaning +3 is the same value as 3. NEGATIVE INTEGERS Are numbers to the left of zero. Are valued less than zero. Express ideas of down or a loss. The sign for a negative integer is (-). This sign is always needed. Opposite Numbers/Integers – are the pairs of integers that have the same absolute value or have the same distance away from zero. ABSOLUTE VALUE The distance of a number from the origin (0) regardless of direction is called absolute value. The absolute value of a number is never negative. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. Examples: I 4 I = 4, +4 is read “ the absolute value of 4 is 4 “ I -3 I = 3, -3 is read “ the absolute value of -3 is 3” - I 3 I = -3, means “ the negative of the absolute value of 3 is -3 “ COMPARING AND ARRANGING INTEGERS Integers can be compared using a number line. As you move to the left along the number line, the integers decrease in value. On the other hand, integers increase in value as you move to the right along the number line. To arrange integers in ascending order is to arrange them from least to greatest. This means that when you use the number line, the smallest the integer is to the left of 0 on the number line. To arrange integers in descending order is to arrange them from greatest to least. This means that when you use the number line, the largest the integer is to the right of 0 on the number line. This is read as “nine is greater than negative 12.” This is read as “negative thirteen is less than negative 5.” This is read as “negative eight is greater than negative 18.” R

What are INTEGERS? Integers are whole numbers that describe opposite ideas in mathematics. Integers can either be negative (-), positive (+) or zero. The integer zero is neutral. It is neither positive nor negative, but is an integer. Integers can be represented on a number line, which can help us understand the value of the integer. POSITIVE INTEGERS Are numbers to the right of zero. Are valued greater than zero. Express ideas of up, a gain or a profit. The sign for a positive integer is (+), however the sign is not always needed. Meaning +3 is the same value as 3. NEGATIVE INTEGERS Are numbers to the left of zero. Are valued less than zero. Express ideas of down or a loss. The sign for a negative integer is (-). This sign is always needed. Opposite Numbers/Integers – are the pairs of integers that have the same absolute value or have the same distance away from zero. ABSOLUTE VALUE The distance of a number from the origin (0) regardless of direction is called absolute value. The absolute value of a number is never negative. The symbol for absolute value is two straight lines surrounding the number or expression for which you wish to indicate absolute value. Examples: I 4 I = 4, +4 is read “ the absolute value of 4 is 4 “ I -3 I = 3, -3 is read “ the absolute value of -3 is 3” - I 3 I = -3, means “ the negative of the absolute value of 3 is -3 “ COMPARING AND ARRANGING INTEGERS Integers can be compared using a number line. As you move to the left along the number line, the integers decrease in value. On the other hand, integers increase in value as you move to the right along the number line. To arrange integers in ascending order is to arrange them from least to greatest. This means that when you use the number line, the smallest the integer is to the left of 0 on the number line. To arrange integers in descending order is to arrange them from greatest to least. This means that when you use the number line, the largest the integer is to the right of 0 on the number line. This is read as “nine is greater than negative 12.” This is read as “negative thirteen is less than negative 5.” This is read as “negative eight is greater than negative 18.”

Absolute Value and Distance in the Complex Plane - Exit Quiz

Find the absolute value and illustrate in a number line

Short Quiz in Math 7 represents the absolute value of a number on a number line as the distance

Covalent Molecules and Compounds Just as an atom is the simplest unit that has the fundamental chemical properties of an element, a molecule is the simplest unit that has the fundamental chemical properties of a covalent compound. Some pure elements exist as covalent molecules. Hydrogen, nitrogen, oxygen, and the halogens occur naturally as the diatomic (“two atoms”) molecules H2, N2, O2, F2, Cl2, Br2, and I2 (part (a) in Figure 3.1.1). Similarly, a few pure elements exist as polyatomic (“many atoms”) molecules, such as elemental phosphorus and sulfur, which occur as P4 and S8 (part (b) in Figure 3.1.1). Each covalent compound is represented by a molecular formula, which gives the atomic symbol for each component element, in a prescribed order, accompanied by a subscript indicating the number of atoms of that element in the molecule. The subscript is written only if the number of atoms is greater than 1. For example, water, with two hydrogen atoms and one oxygen atom per molecule, is written as H2O. Similarly, carbon dioxide, which contains one carbon atom and two oxygen atoms in each molecule, is written as CO2. Covalent compounds that predominantly contain carbon and hydrogen are called organic compounds. The convention for representing the formulas of organic compounds is to write carbon first, followed by hydrogen and then any other elements in alphabetical order (e.g., CH4O is methyl alcohol, a fuel). Compounds that consist primarily of elements other than carbon and hydrogen are called inorganic compounds; they include both covalent and ionic compounds. In inorganic compounds, the component elements are listed beginning with the one farthest to the left in the periodic table, as in CO2 or SF6. Those in the same group are listed beginning with the lower element and working up, as in ClF. By convention, however, when an inorganic compound contains both hydrogen and an element from groups 13–15, hydrogen is usually listed last in the formula. Examples are ammonia (NH3) and silane (SiH4). Compounds such as water, whose compositions were established long before this convention was adopted, are always written with hydrogen first: Water is always written as H2O, not OH2. The conventions for inorganic acids, such as hydrochloric acid (HCl) and sulfuric acid (H2SO4), are described elswhere. Note! For organic compounds: write C first, then H, and then the other elements in alphabetical order. For molecular inorganic compounds: start with the element at far left in the periodic table; list elements in same group beginning with the lower element and working up. Write the molecular formula of each compound. a. The phosphorus-sulfur compound that is responsible for the ignition of so-called strike anywhere matches has 4 phosphorus atoms and 3 sulfur atoms per molecule. b. Ethyl alcohol, the alcohol of alcoholic beverages, has 1 oxygen atom, 2 carbon atoms, and 6 hydrogen atoms per molecule. c. Freon-11, once widely used in automobile air conditioners and implicated in damage to the ozone layer, has 1 carbon atom, 3 chlorine atoms, and 1 fluorine atom per molecule. Solution: a. • A The molecule has 4 phosphorus atoms and 3 sulfur atoms. Because the compound does not contain mostly carbon and hydrogen, it is inorganic. • B Phosphorus is in group 15, and sulfur is in group 16. Because phosphorus is to the left of sulfur, it is written first. • C Writing the number of each kind of atom as a right-hand subscript gives P4S3 as the molecular formula. b. • A Ethyl alcohol contains predominantly carbon and hydrogen, so it is an organic compound. • B The formula for an organic compound is written with the number of carbon atoms first, the number of hydrogen atoms next, and the other atoms in alphabetical order: CHO. • C Adding subscripts gives the molecular formula C2H6O. c. • A Freon-11 contains carbon, chlorine, and fluorine. It can be viewed as either an inorganic compound or an organic compound (in which fluorine has replaced hydrogen). The formula for Freon-11 can therefore be written using either of the two conventions. • B According to the convention for inorganic compounds, carbon is written first because it is farther left in the periodic table. Fluorine and chlorine are in the same group, so they are listed beginning with the lower element and working up: CClF. Adding subscripts gives the molecular formula CCl3F. • C We obtain the same formula for Freon-11 using the convention for organic compounds. The number of carbon atoms is written first, followed by the number of hydrogen atoms (zero) and then the other elements in alphabetical order, also giving CCl3F. Write the molecular formula for each compound. a. Nitrous oxide, also called “laughing gas,” has 2 nitrogen atoms and 1 oxygen atom per molecule. Nitrous oxide is used as a mild anesthetic for minor surgery and as the propellant in cans of whipped cream. b. Sucrose, also known as cane sugar, has 12 carbon atoms, 11 oxygen atoms, and 22 hydrogen atoms. c. Sulfur hexafluoride, a gas used to pressurize “unpressurized” tennis balls and as a coolant in nuclear reactors, has 6 fluorine atoms and 1 sulfur atom per molecule. Answer: a. N2O b. C12H22O11 c. SF6. Ionic Compounds The substances described in the preceding discussion are composed of molecules that are electrically neutral; that is, the number of positively-charged protons in the nucleus is equal to the number of negatively-charged electrons. In contrast, ions are atoms or assemblies of atoms that have a net electrical charge. Ions that contain fewer electrons than protons have a net positive charge and are called cations. Conversely, ions that contain more electrons than protons have a net negative charge and are called anions. Ionic compounds contain both cations and anions in a ratio that results in no net electrical charge. Note! Ionic compounds contain both cations and anions in a ratio that results in zero electrical charge.An ionic compound that contains only two elements, one present as a cation and one as an anion, is called a binary ionic compound. One example is MgCl2, a coagulant used in the preparation of tofu from soybeans. For binary ionic compounds, the subscripts in the empirical formula can also be obtained by crossing charges: use the absolute value of the charge on one ion as the subscript for the other ion. This method is shown schematically as follows: Crossing charges. One method for obtaining subscripts in the empirical formula is by crossing charges. When crossing charges, it is sometimes necessary to reduce the subscripts to their simplest ratio to write the empirical formula. Consider, for example, the compound formed by Mg2+ and O2−. Using the absolute values of the charges on the ions as subscripts gives the formula Mg2O2:Polyatomic Ions Polyatomic ions are groups of atoms that bear net electrical charges, although the atoms in a polyatomic ion are held together by the same covalent bonds that hold atoms together in molecules. Just as there are many more kinds of molecules than simple elements, there are many more kinds of polyatomic ions than monatomic ions. Two examples of polyatomic cations are the ammonium (NH4+) and the methylammonium (CH3NH3+) ions. P. The method used to predict the empirical formulas for ionic compounds that contain monatomic ions can also be used for compounds that contain polyatomic ions. The overall charge on the cations must balance the overall charge on the anions in the formula unit. Thus, K+ and NO3− ions combine in a 1:1 ratio to form KNO3 (potassium nitrate or saltpeter), a major ingredient in black gunpowder. Similarly, Ca2+ and SO42− form CaSO4 (calcium sulfate), which combines with varying amounts of water to form gypsum and plaster of Paris. The polyatomic ions NH4+ and NO3− form NH4NO3 (ammonium nitrate), a widely used fertilizer and, in the wrong hands, an explosive. One example of a compound in which the ions have charges of different magnitudes is calcium phosphate, which is composed of Ca2+ and PO43− ions; it is a major component of bones. The compound is electrically neutral because the ions combine in a ratio of three Ca2+ ions [3(+2) = +6] for every two ions [2(−3) = −6], giving an empirical formula of Ca3(PO4)2; the parentheses around PO4 in the empirical formula indicate that it is a polyatomic ion. Writing the formula for calcium phosphate as Ca3P2O8 gives the correct number of each atom in the formula unit, but it obscures the fact that the compound contains readily identifiable PO43− ions.Summary • There are two fundamentally different kinds of chemical bonds (covalent and ionic) that cause substances to have very different properties. • The composition of a compound is represented by an empirical or molecular formula, each consisting of at least one formula unit.Contributors The atoms in chemical compounds are held together by attractive electrostatic interactions known as chemical bonds. Ionic compounds contain positively and negatively charged ions in a ratio that results in an overall charge of zero. The ions are held together in a regular spatial arrangement by electrostatic forces. Most covalent compounds consist of molecules, groups of atoms in which one or more pairs of electrons are shared by at least two atoms to form a covalent bond. The atoms in molecules are held together by the electrostatic attraction between the positively charged nuclei of the bonded atoms and the negatively charged electrons shared by the nuclei. The molecular formula of a covalent compound gives the types and numbers of atoms present. Compounds that contain predominantly carbon and hydrogen are called organic compounds, whereas compounds that consist primarily of elements other than carbon and hydrogen are inorganic compounds. Diatomic molecules contain two atoms, and polyatomic molecules contain more than two. A structural formula indicates the composition and approximate structure and shape of a molecule. Single bonds, double bonds, and triple bonds are covalent bonds in which one, two, and three pairs of electrons, respectively, are shared between two bonded atoms. Atoms or groups of atoms that possess a net electrical charge are called ions; they can have either a positive charge (cations) or a negative charge (anions). Ions can consist of one atom (monatomic ions) or several (polyatomic ions). The charges on monatomic ions of most main group elements can be predicted from the location of the element in the periodic table. Ionic compounds usually form hard crystalline solids with high melting points. Covalent molecular compounds, in contrast, consist of discrete molecules held together by weak intermolecular forces and can be gases, liquids, or solids at room temperature and pressure. An empirical formula gives the relative numbers of atoms of the elements in a compound, reduced to the lowest whole numbers. The formula unit is the absolute grouping represented by the empirical formula of a compound, either ionic or covalent. Empirical formulas are particularly useful for describing the composition of ionic compounds, which do not contain readily identifiable molecules. Some ionic compounds occur as hydrates, which contain specific ratios of loosely bound water molecules called waters of hydration.

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