Ariannis Hines 8-1
Algebra 2 Midterm Study Guide
Section 1.7, Introduction to Solving Inequalities
Solving and graphing one-variable inequalities
Problem: 16x-9 < 23x+11
Steps:
-16x+16x-9 23x+11-16x
(note: the smaller variable should be subtracted from both sides of the inequality first.)
-9 7x+11
(note: when subtracting or adding a number to an inequality, you never change or alter the inequality sign.)
-11-9 7x+11-11
-19 7x
-197< 7x7
(note: once you’ve simplified the inequality so that there’s one variable on one side of the inequality, divide both sides by the number attached to the variable, in this case, 7.)
Solution:
Graphing the Solution:
-257 0
(note: if the solution has a greater-than-or-equal-to, or a less-than-equal-to sign (≥,≤), the circle in the graph has to be colored in. Otherwise, like in this case, you leave the circle uncolored.)
(note: since x is greater than -257, it is greater than any number larger than -257. Therefore, when graphing the inequality, you draw an arrow stemming from your circle, which represents x, and draw it going left or right, depending on the inequality. Since x is greater than -257, you draw the arrow going to the right.)
Problem: -6x-5 ≥ 13
Steps:
5-6x-5 ≥ 13+5
-6x ≥ 18
-6x-6≥18-6
(note: when you divide or multiply a inequality by a negative number, you change the sign. If the sign was originally less-than, you would change it to greater-than and vice versa.)
Solution:
Graphing the Solution:
-3 0
Solving and graphing two-variable inequalities
-To solve a compound inequality involving and, find the values of the variable that satisfy both inequalities.
Problem: 2x+1≥3 and 3x-4≤17
Steps:
2x+1≥3 and 3x-4≤17
-1+2x+1≥3-1 4+3x-4≤17+4
2x≥2 3x≤21
2x2≥22 3x3≤213
Solution:
(note: with an and inequality, you can combine the two answers by making a compound inequality.)
Graphing the Solution:
1 0 7
(note: the solution is all values of x between 1 and 7 inclusive.)
-When you solve a compound inequality involving or, find those values of the variable that satisfy at least one of the inequalities.
Problem: 5x+1>21 or 3x+2<-1
Steps:
5x+1>21 or 3x+2<-1
-1+5x+1>21-1 -2+3x+2<-1-2
5x>20 3x<-3
5x5205 3x3-33
Solution:
Graphing the Solution:
-1 0 4
(note: the solution is all values of x less than -1 or greater than 4.)