Key Concepts
Translating Word Problems Into Expressions
| Addition | [latex]+[/latex] | sum, total, together, more than, increased by, … |
| Subtraction | [latex]-[/latex] | difference, less than, decreased by, … |
| Multiplication | [latex]\times[/latex] | product, double, half, percent of, … |
| Equality | [latex]=[/latex] | is, is the same as, is not different from, … |
Inequality Symbols
Symbol |
In Words |
Example |
| [latex]\neq[/latex] | Not equal to | [latex]3 \neq 5[/latex] |
| [latex]>[/latex] | Greater than | [latex]5>3[/latex] |
| [latex]<[/latex] | Less than | [latex]2<7[/latex] |
| [latex]\geq[/latex] | Greater than or equal to | [latex]5 \geq 5[/latex] |
| [latex]\leq[/latex] | Less than or equal to | [latex]4 \leq 7[/latex] |
The inequality [latex]a<b[/latex] can also be written [latex]b>a[/latex].
Graphing an Inequality
- Locate the endpoint of the inequality on the number line. Draw a closed (shaded) circle if the inequality includes the endpoint [latex](\geq \mathrm{or} \leq)[/latex]. Draw an open circle for a strict inequality ([latex]< \mathrm{or} >[/latex]).
- Shade the portion of the number line which satisfies the stated inequality.
Translating Verbal Statements Involving Inequalities
Verbal Statement: x is… |
Mathematical Statement |
| greater than or equal to [latex]k[/latex] at least [latex]k[/latex] not more than [latex]k[/latex] |
[latex]x \geq k[/latex] |
| less than or equal to [latex]k[/latex] at most [latex]k[/latex] not more than [latex]k[/latex] |
[latex]x \leq k[/latex] |
| less than [latex]k[/latex] below [latex]k[/latex] fewer than [latex]k[/latex] |
[latex]x<k[/latex] |
| greater than [latex]k[/latex] more than [latex]k[/latex] above [latex]k[/latex] |
[latex]x>k[/latex] |
| equal to [latex]k[/latex] is [latex]k[/latex] exactly [latex]k[/latex] the same as [latex]k[/latex] |
[latex]x=k[/latex] |
| not equal to [latex]k[/latex] not [latex]k[/latex] different from [latex]k[/latex] |
[latex]x \neq k[/latex] |
To convert a number in scientific notation to standard notation, move the decimal point [latex]n[/latex] places to the write if [latex]n[/latex] is positive, or [latex]|n|[/latex] places to the left if [latex]n[/latex] is negative. Add zeros as needed.
To write a number in scientific notation, move the decimal point to the right of the first nonzero digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places [latex]n[/latex] that you moved the decimal point. Multiply the decimal number by 10 raised to a power of [latex]n[/latex]. If you moved the decimal left as in a very large number, [latex]n[/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[/latex] is negative.
Glossary
constant: number whose value does not change
inequality: compares two expressions, identifying one as more, less, or simply different than the other
inference: drawing reliable conclusions about the population on the basis of what we’ve discovered in our sample
probability: a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; also described as long-term relative frequency. Probabilities are between 0 and 1, inclusive.
real number line: a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.
scientific notation: number is written in the form [latex]a \times 10^{n}[/latex] where [latex]1 \leq |a| < 10[/latex]
solution set: values of a variable which make a statement true
variable: a quantity that may change value, or whose value we don’t know
