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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. IDENTITIES AND CONTRADICTIONS 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 ADDITION AND MULTIPLICATION 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 Reading: 1.6 Linear Modeling