### 1.5 Linear Equations and Inequalities

```1.5 LINEAR EQUATIONS AND
INEQUALITIES
QUIZ

Tell true or false of the following statement:
If c < 0, a < b, then ac > bc.
LINEAR EQUATION
A linear equation in one variable is an equation
that can be written in the form:
ax+b=0, a≠0
LINEAR EQUATION
Addition and Multiplication Properties of Equality
1, if a=b, then a+c=b+c for any c∈R
2, If c≠0, a=b, then ac=bc, a/c=b/c.
SOLVE A LINEAR EQUATION ANALYTICALLY

Find the zero of the function f.
1, f(x)=-3x-12
2, f(x)=-4(2x-3)+8(2x+1)
SOLVE A LINEAR EQUATION BY GRAPH

To solve the equation f(x)=g(x) graphically, graph
y1 =f(x) and y2=g(x)
The x-coordinate of any point of intersection of
the two graphs is a solution of the equation.
SOLVE A LINEAR EQUATION BY GRAPH

X-Intercept Method of Graphical Solution
To solve the equation f(x)=g(x) graphically, graph
y =f(x) -g(x)=F(x)
Any x-intercept of the graph of y = F(x) is a
solution of the equation.
Recall: x-intercept is the zero of the linear
function.
Identity: an equation that is true for all values in
the domain of its variables.
ex: 5(x+1)=5x+5
Contradiction: an equation that has no solution.
ex: x+1=x+3
INEQUALITIES IN ONE VARIABLE

Notation:
a<b
a>b
a≤b
a≥b
a is less than b
a is greater
than b
a is less or
equal to b
a is greater or
equal to b
PROPERTIES OF INEQUALITY

For real numbers a, b and c
1, if a < b, then a + c < b + c.
2, If a < b, c > 0, then ac < bc
3, if a < b, c < 0, then ac > bc
Slimier properties exist for >, ≤ and ≥
LINEAR INEQUALITY IN ONE VARIABLE

A linear inequality in one variable is an
inequality that can be written in one of the
following forms, where a ≠ 0:
ax+b>0, ax+b<0, ax+b ≥0, ax+b
≤0
SOLVING LINEAR INEQUALITIES

Exercise:
1, 10x+5-7x ≥8(x+2)+4
2, (2x+3)/5-(3x-1)/2<(4x+7)/2
GRAPHICAL APPROACHES
f(x)
f(x) < g(x)
g(x)
f(x) > g(x)
f(x) ≤ g(x)
f(x) ≥ g(x)
GRAPHICAL APPROACHES

X-Intercept Method of Solution of a linear
Inequality
The solution set of F(x)>0 is the set of all real
numbers x such that the graph of F is above the
x-axis. The solution set of F(x)<0 is the set of all
real numbers x such that the graph of F is below
the x-axis.
THREE – PART INEQUALITIES

Three – Part inequalities have the form of :
g(x) < f(x) <h(x)
g(x) ≤ f(x) <h(x)
g(x) < f(x) ≤ h(x)
g(x) ≤ f(x) ≤ h(x)
ex: -3< 2x+1 < 2
x+1 < 3x+4 < 2x+6
HOMEWORK

PG. 57: 25-100 (M5)

KEY: 30,70,85
