1. Solving One-Step Inequalities
Algebra 1 ~ Chapter 6.1 and 6.2
Solving One-Step Inequalities

2. Recall that statements with greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) are inequalities.
Solving one-step inequalities is much like solving one-step equations.
To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations.

4. The solution set is {all numbers
Ex. 1 - Solve the inequality and graph the solutions.
x + 12 < 20
–12 –12
The solution set is {all numbers
less than 8}.
x < 8
The circle at 8 is open. This shows that 8 is NOT included in the inequality.
–10 –8 –6 –4 –2 2 4 6 8 10
The heavy arrow pointing to the left shows that the inequality includes all #s less 8.
Check your solution??

5. Ex. 2 - Solve the inequality and graph the solutions.
d > –2
Example check:
d = 0
d – 5 > -7
0 – 5 > -7
-5 > -7
TRUE!
Example check:
d = -6
d – 5 > -7
> -7
-11 > -7
FALSE!
–10 –8 –6 –4 –2 2 4 6 8 10

6. Ex. 3 – Solving an Inequality with Variables on both sides
Solve x – 4 ≤ 13x
-12x x
-4 ≤ 1x
x ≥ -4

8. x > –6 Ex. 4 - Solve the inequality and graph the solutions. CHECK
7(0) > -42
0 > -42
TRUE!
7x > –42
CHECK
7x > -42
7(-10)>-42
-70 > -42
FALSE!
x > –6
–10 –8 –6 –4 –2 2 4 6 8 10

9. Ex. 5 - Solve the inequality and graph the solutions.
Check:
3(2) ≤ 3
6 ≤ m
(or m ≥ 6) 2 4 6 8 10 12 14 16 18 20

10. Ex. 6 - Solve the inequality and graph the solutions.
Since r is multiplied by , multiply both sides by the reciprocal of .
r < 16 2 4 6 8 10 12 14 16 18 20

11. What happens when you multiply or divide both sides of an inequality by a negative number?
Look at the number line below. –b –a a b
a < b
–a –b
You can tell from the number line that –a > –b.
Multiply both sides by –1.
b > –a
–b a
Multiply both sides by –1.
You can tell from the number line that –b < a.
>
<
Notice that when you multiply (or divide) both sides of an inequality by a negative number, you must REVERSE the inequality symbol.

12. A real number example… 3 < 8 -1(3) ? -1(8) -3 -8 >
Start off with the # 8 is greater than the # 3. TRUE!
3 < 8
-1(3) ? -1(8)
If I multiply both sides by a negative # (-1 in this case)…
In order to keep this inequality TRUE, what must I do to the inequality symbol?
>
REVERSE or FLIP the inequality sign!!

14. Caution!
Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

15. Ex. 7 - Solve the inequality and graph the solutions.
Since x is multiplied by –12, divide both sides by –12. Change > to <.
x < –7
CHECK
-12x > 84
-12(-12) > 84
144 > 84
TRUE!
–10 –8 –6 –4 –2 2 4 6
–12
–14 –7

16. Ex. 8 - Solve the inequality and graph the solutions.
Since x is divided by –3, multiply both sides by –3. Change to .
24  x
(or x  24) 16 18 20 22 24 10 14 26 28 30 12

17. Multiply both sides by –1 to make x positive. Change  to .
Ex Solve the inequality and graph the solutions. Check your answer.
10 ≥ –x
Multiply both sides by –1 to make x positive. Change  to .
–1(10) ≤ –1(–x)
CHECK
10 ≥ –x
10 ≥ -(4)
10 ≥ -4
TRUE!
–10 ≤ x (or even better x ≥ -10)
–10 –8 –6 –4 –2 2 4 6 8 10

18. Ex. 10 – Define a variable and write an inequality for each problem
Ex. 10 – Define a variable and write an inequality for each problem. You do not need to solve the inequality.
a.) A number decreased by 8 is at most 14.
b.) A number plus 7 is greater than 2.
c.) Half of a number is at least 26.
n – 8 ≤ 14
n + 7 > 2
½n ≥ 26

19. Lesson Review Solve each inequality and graph the solutions.
1. 13 < x + 7
x > 6
2. –6 + h ≥ 15
h ≥ 21 3.
4. –5x ≥ 30
x > 20
x ≤ –6

20. Assignment Worksheet 6-1 & 6-2 (Front and Back)
Pages #’s (evens), (evens)
Pages #’s (evens), (evens)