1-3, 1-4, and 1-5 Expressions, Equations, and Inequalities
Quick Review
You evaluate an algebraic expression by substituting numbers for the variables. You simplify an algebraic expression by combining like terms. To find the solution of an equation or inequality, use the properties of equality or inequality. Some equations and inequalities are true for all real numbers, and some have no solution.equalities are true for all real numbers, and some have no solution.
Example
Evaluate
3
(
x
−
4
)
+
2
x
−
x
2
for
x
=
6
.
3 open x minus 4 close plus 2 x minus , x squared , for , x equals 6 .
3
(
6
−
4
)
+
2
(
6
)
−
6
2
Substitute.
=
3
(
2
)
+
2
(
6
)
−
6
2
Simplify inside parentheses.
=
6
+
12
−
36
Multiply.
=
18
−
36
Add.
=
−
18
Subtract.
table with 5 rows and 3 columns , row1 column 1 , , column 2 3 open 6 minus 4 close plus 2 open 6 close minus , 6 squared , column 3 cap substitute. , row2 column 1 , , column 2 equals 3 open 2 close plus 2 open 6 close minus , 6 squared , column 3 cap simplify inside parentheses. , row3 column 1 , , column 2 equals 6 plus 12 minus 36 , column 3 cap multiply. , row4 column 1 , , column 2 equals 18 minus 36 , column 3 cap add. , row5 column 1 , , column 2 equals negative 18 , column 3 cap subtract. , end table
Exercises
- Evaluate
3
t
(
t
+
2
)
−
3
t
2
for
t
=
19
.
3 t open t plus 2 close minus 3 , t squared , for , t equals 19 .
- Simplify
−
(
3
a
−
2
b
)
−
3
(
−
a
−
b
)
.
negative open 3 eh minus 2 b close minus 3 open negative eh minus b close .
Solve each equation. Check your answer.
-
2
x
−
5
=
17
2 x minus 5 equals 17
-
3
(
x
+
1
)
=
9
+
2
x
3 open x plus 1 close equals 9 plus 2 x
Solve each inequality. Graph the solution.
-
4
−
5
z
≥
2
4 minus 5 z greater than or equal to 2
-
2
(
5
−
3
x
)
<
x
−
4
(
3
−
x
)
2 open 5 minus 3 x close less than x minus 4 open 3 minus x close
Solve each compound inequality. Graph the solution.
-
10
≥
7
+
3
x
and
9
−
4
x
≤
1
10 greater than or equal to 7 plus 3 x , and , 9 minus 4 x less than or equal to 1
-
3
≥
2
x
or
x
−
4
>
2
3 greater than or equal to 2 x , or , x minus 4 greater than 2
Write an equation to solve the problem.
-
Geometry The length and width of a rectangle are in the ratio 5 : 3. The perimeter of the rectangle is 32 cm. Find the length and width.
1-6 Absolute Value Equations and Inequalities
Quick Review
To rewrite an equation or inequality that involves the absolute value of an algebraic expression, you must consider both cases of the definition of absolute value.
Example
Solve
|
3
x
−
5
|
=
4
+
2
x
.
vertical line 3 x minus 5 vertical line equals 4 plus 2 x . Check for extraneous solutions.
Image Long Description
Check
|
3
(
9
)
−
5
|
=
?
4
+
2
(
9
)
|
27
−
5
|
=
?
22
|
22
|
=
22
*
|
3
(
1
5
)
−
5
|
=
?
4
+
2
(
1
5
)
|
3
5
−
5
|
=
?
4
+
2
5
|
−
22
5
|
=
22
5
*
table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 2 columns , row1 column 1 , absolute value of . 3 , open 9 close , minus 5 , end absolute value , , column 2 modified equals with question mark above , 4 plus 2 , open 9 close , row2 column 1 , absolute value of , 27 minus 5 , end absolute value , , column 2 modified equals with question mark above , 22 , row3 column 1 , absolute value of 22 , , column 2 equals 22 times , end table , column 2 table with 3 rows and 2 columns , row1 column 1 , absolute value of . 3 . open , 1 fifth , close . minus 5 , end absolute value , , column 2 modified equals with question mark above , 4 plus 2 . open , 1 fifth , close , row2 column 1 , absolute value of . 3 fifths , minus 5 , end absolute value , , column 2 modified equals with question mark above , 4 plus , 2 fifths , row3 column 1 , absolute value of , negative , 22 over 5 , end absolute value , , column 2 equals , 22 over 5 , times , end table , end table
Exercises
Solve each equation. Check for extraneous solutions.
-
|
2
x
+
8
|
=
3
x
+
7
vertical line 2 x plus 8 vertical line equals 3 x plus 7
-
|
x
−
4
|
+
3
=
1
vertical line x minus 4 vertical line plus 3 equals 1
-
3
|
x
+
10
|
=
6
3 vertical line x plus 10 vertical line equals 6
-
2
|
x
−
7
|
=
x
−
8
2 vertical line x minus 7 vertical line equals x minus 8
Solve each inequality. Graph the solution.
-
|
3
x
−
2
|
+
4
≤
7
vertical line 3 x minus 2 vertical line plus 4 less than or equal to 7
-
4
|
y
−
9
|
>
36
4 vertical line y minus 9 vertical line greater than 36
-
|
7
x
|
+
3
≤
21
vertical line 7 x vertical line plus 3 less than or equal to 21
-
1
2
|
x
+
2
|
>
6
1 half , vertical line x plus 2 vertical line greater than 6
- The specification for a length x is 43.6 cm with a tolerance of 0.1 cm. Write the specification as an absolute value inequality.
