Summary: Review

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]aa[/latex].

Graphing an Inequality

  1. 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]).
  2. 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
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