FirstDegree Equations and Inequalities

1
First-Degree Equations and Inequalities
2
Learning Objectives

• Defining first-degree equations and inequalities.
• Solving first-degree equations.
• Graphs and first-degree equations.
• Formulas and literal equations.
• Applications.
• Solving first-degree inequalities.

3
Definitions

• Equation an equation is a mathematical statement
  that two expressions are equal. (Ex x2-2x10).
• Identity an equation that is always true is
  called an identity. (Ex (-x)2 x2).
• Conditional an equation that is sometimes true
  is called a conditional equation. (Ex
  x2-2x10).
• Contradiction an equation that is never true is
  called a contradiction. (Ex x2 2x 1 -5).
• Solution or root a value of the variable that
  results in a true equation is called a solution
  or root of the equation. It is said to satisfy
  the equation.
• Solution set the set of all the solutions of an
  equation is called the solution set of the
  equation.

4
Definitions

• Equivalent equations equations which have
  exactly the same solution set are called
  equivalent equations.(Ex (x-1)2 0 and x2 2x
  1 0).
• First-degree equation a first-degree equation is
  an equation that can be written in the form axb
  0, where a and b are constants and a ? 0.(Ex
  2x 3 0).

5
Definitions

• Inequalities expressions that utilize the
  relations lt, ?, gt or ? are called
  inequalities.(Ex (x-1)2 gt 0).
• Solution any element of the replacement set
  (domain) of the variable for which the inequality
  is true is called a solution.(Ex x -4 is a
  solution of 2x 3 lt -2).
• Solution set the set of all the solutions of an
  inequality is called the solution set of the
  inequality.

6
Definitions

• Equivalent inequalities inequalities which have
  exactly the same solution set are called
  equivalent inequalities.(Ex (x-1)2 gt 0 and x2
  2x 1 gt 0).
• First-degree inequality a first-degree
  inequality is an inequality that can be written
  in the form axb gt 0, where a and b are constants
  and a ? 0. (Ex 2x 3 lt 0).

7
Solving a First-Degree Equation

• Some useful properties
• Addition property of equalityif a b, then a
  q b q
• Multiplication property of equalityif a b,
  then aq bq provided that q ? 0. Example 3x
  4 16 3x 4 4 16 4
  (addition property of equality) 3x
  12 (combine like terms) 3x / 3 12 / 3
  (multiplication property of
  equality) x 4 (simplification)4 is the
  solution or root of the given equality. 4 is
  the solution set.

8
Solving a First-Degree Equation

• Solve 4 (t-1) - t t 2 4t 4 t t
  2 (distribute) 3t 4 t 2 (combine) 3t
  4 t t 2 t (addition property) 2t 4
  -2 (combine) 2t 4 4 -2 4 (addition
  property) 2t 2 (combine) 2t/2 2/2
  (multiplication property) t 1 (simplify)Th
  e solution is t 1.
• Check t 1 satisfies the equality.

9
Solving a First-Degree Equation

• Solve n/3 4 (n-1)/4 (n/3 4) 12
  ((n-1) / 4) 12 (multiply by LCD) 4n 48 3
  (n-1) (simplify) 4n 48 3n -
  3 (distribute) 4n 48 3n 3n 3 - 3n
  (subtract 3n) n 48 -3 (combine) n 48 -
  48 -3 -48 (subtract 48) n
  -51 (combine)The solution is n -51.
• Check n -51 satisfies the equality.

10
Solving a First-Degree Equation

• Theorem when there is a solution to first-degree
  equation, it is unique.Demonstration let ax
  b 0 be a first-degree equation. Let r be its
  solution. We want to show that ar b as
  b ar b b as b b (subtract b) ar
  as (simplify) ar/a as/a (divide by a) r
  s (simplify) QEDNote QED
  stands for Quod Erat Demonstrandum (This has been
  demonstrated)

11
Graphs of First-Degree Equations

• First-degree equation with two variables a
  first-degree equation with two variables is an
  equation of the form y ax b. A solution is a
  pair of values.
• Each pair of variables is represented as (x,y) in
  alphabetical order.
• There are infinitely many ordered pairs
  satisfying the equation.Ex y 2x 5
  (0,5), (1,7), (2, 9), are solutions.
• This is why it is best to represent the solutions
  by a graph than to enumerate the solution set.

12
Graphs of First-Degree Equations

• Abscissa, ordinate in (x,y), x is the abscissa
  and y is the ordinate.
• Cartesian coordinate system developed by René
  Descartes in the 17th century, the cartesian
  coordinate system is composed of an x-axis and a
  y-axis, perpendicular to one another at a point
  called the origin. The coordinate axes separate
  the plane into four regions called quadrants, and
  numbered in Roman numerals.
• An ordered pair is represented by its graph a
  point with coordinates (x-coordinate,
  y-coordinate). Coordinates of the origin (0,0).

13
Graphs of First-Degree Equations

• Example plot the solution set of the equation
  yx-2.

14
Graphs of First-Degree Equations

• Maple commands for plotting
• plot(function, x-range, y-range (optional) , )
• Examples plot(x-2, x-6..6)plot(x2, x)
  will take the x-range as
  10..10plot(sin(x), x-Pi..Pi)plot(sin,
  -Pi..Pi)plot(sin(x), x0..infinity)
• Multiple plotsplot(sin(x), x-x3/6, x0..2,
  colorred, blue, stylepoint, line)
• Point plotsl n, sin(n) n1..10plot(l,
  x0..15, stylepoint, symbolcircle)

15
Graphs of First-Degree Equations

• Theorem1 the graph of an equation of the form ax
  by c, where a and b are not both zero, is a
  straight line. Conversely, every straight line is
  the graph of an equation of the form ax by
  c.
• An equation of the form axby c is called a
  linear equation for this reason.
• The form axby c is called the standard form of
  the equation.
• It is sufficient to know two particular points to
  graph a line for instance the x-intercept
  (intersection with the x-axis) and the
  y-intercept (intersection with the y-axis).
• The graph of xc is a vertical line, the graph of
  yc is a horizontal line.

16
Graphs of First-Degree Equations

• Example graph x-2y 6. We can transform it into
  y x/2 3 (for Maple). X-intercept y0, x6.
  Y-intercept x0, y-3.

17
Formulas and Literal Equations

• Literal equations an equation that contains
  more than one variable is called a literal
  equation.
• Formula a literal equation that expresses how
  quantities encountered in practical applications
  are related is called a formula.(Ex p 2 ? r,
  a ½ b h).
• A literal equation is said to be solved
  explicitly for a variable if that variable is
  isolated on one side of the equation.

18
Formulas and Literal Equations

• Example

19
Applications

• Guidelines for solving real-world problems
• Read the problem carefully !
• Draw a diagram or picture whenever possible.
• Identify the unknown quantity/quantities and use
  a variable name for each.
• Determine how the known and unknown quantities
  are related.
• Write an equation relating the known and unknown
  quantities.
• Solve the equation.
• Answer the question that was asked.
• Check the answer in the statement of the problem.
• Ratio the quotient of two quantities, a/b, is
  called a ratio.
• Proportion A statement that two ratios are
  equal, a/b c/d, is called a proportion.

20
Applications

• Example
• Let x be the number of peaches left.
• We can write a proportion 90 / 30 x / 70.
• Thus x 90 . 70 / 30 3 . 70 210.
• There are 210 peaches left.
• Check 30 (210 90) 30 300 90.

21
Solving First-Degree Inequalities

• Properties of inequalitiesif a lt b, then for
  real numbers a, b and q
• aq lt bq and a-q lt b-q (addition/subtraction)
• If q gt 0, aq lt bq and a/q lt b/q (product/division
  by positive number)
• If q lt 0, aq gt bq and a/q gt b/q (product/division
  by negative number)
• The above properties are also valid for gt, ? and
  ?.

22
Solving First-Degree Inequalities

• Example5(n-2) 3(n4) ? n 205n 10 3n 12 ?
  n 20 (distribute)2n 22 ? n
  20 (combine)2n 22-n ? n 20 -n (subtract
  n)n 22 ? 20 (combine)n 22 22 ? 20
  22 (add 22)n ? 2 (simplify)The solution
  set is the interval 2, ?).

23
Graphs of First-Degree Inequalities

• Graph the equation ax by c. Then shade a
  half-plane solution of an inequality (on either
  side of a line in a plane).
• Represent the line as plain if the points on the
  line are included in the solution set, and as
  dotted if the points on the line are not included
  in the solution set.

24
Graphs of First-Degree Inequalities

• Example x gt ½ y 3This inequality is
  equivalent to y lt 2x-6. Corresponds on the graph
  to half-plane below and including the dotted line.

25
Applications Involving Inequalities

• A retiree requires an annual income of at least
  1500 from an investment that earns interest at
  7.5 per year. What is the smallest amount the
  retiree must invest in order to achieve the
  desired return ?I Prt (I interest,
  Pprincipal, rrate, ttime) I Prt ??
  1500 P ?? 1500 / rt 1500 / .075 . 1 P ??
  20,000The minimum amount the retiree must invest
  is 20,000.

26
Absolute-value Equations and Inequalities

• Property if c ? 0, q c is equivalent to q
  c or q -c.
• Example solve 2n 7 3 2n 7 3 ?
  2n 7 ? 0 and 2n 7 3 (1) or 2n 7
  ? 0 and 2n 7 -3 (2)(1) ? n ? 7/2 and n 5,
  so n 5 is a solution.(2) ? n ? 7/2 and n 2
  4/2 which is smaller than 7/2, so n
  2 is another solution.The solution set is 2, 5
  .