Algebraic Expressions, Equations, and Inequalities
An algebraic expression is any expression built up from numbers, variables, and operations. Note that an expression is not an equation; an equation contains an equals sign. An algebraic expression contains terms, which are numbers and/or terms combined together. Terms are separated by operations.
Algebraic expressions can be simplified by combining like terms. Combining like terms involves combining terms with the same variable and combining constants, which have no variable.
Given the expression 7m + 14m - 3n + 11n - 5m,
Step 1: Organize the expression by bringing terms with the same variable next to each other.
Our expression becomes 7m + 14m - 5m - 3n + 11n.
Step 2: Combine the coefficients of like terms.
Our expression becomes (7 + 14 - 5)m - (3 - 11)n.
Step 3: Find the values of the new coefficients.
Our expression becomes 16m - (-8)n = 16m + 8n.
Conclusion: 7m + 14m - 3n + 11n - 5m has been simplified down to 16m + 8n.
Example 1. Write an algebraic expression for one-fourth the sum of x and y minus the product of six and z.
Let's split the problem into two parts:
1) one-fourth the sum of x and y ---> one-fourth times (x + y) ---> (x + y)/4
2) the product of six and z ---> six times z ---> 6z
The word minus means we subtract 2) from 1), so our final expressions is (x + y)/4 - 6z.
Expressions are very useful in that we can use them to model real world situations, just like when writing equations based on a word problem.
Example 2. Amy bought two apples and five bananas. Let a represent the cost of one apple and b represent the cost of one banana. Write an algebraic expression for the total amount of money Amy spent on the fruit.
Amy bought two apples, so the cost of both apples combined is 2a.
Amy bought five oranges, so the cost of the five oranges is 5b.
The total amount Amy spent is modeled by the expression 2a + 5b.
An algebraic equation is the equality of two algebraic expressions. The equality of two expressions is denoted by an equals sign, "=". We often use equations to solve for an unknown variable. We can solve for a variable by combining like terms and moving the variable term to one side of the equation with the known terms on the other side.
Examples of algebraic equations include:
a = b
2x + 5 = 14
ax^2 + (b-4)x + c = 0
13m - 45n = 3p/4q
5abcd + e = 6mnp - qr
Example 3. Solve for x in the equation 4x - 7 = 13.
To solve for x, we want to isolate x to one side of the equation by manipulating the rest of the terms, which are constants. We first add 7 to both sides of the equation to cancel out 7 on the left side.
We have 4x - 7 = 13
--> 4x - 7 + 7 = 13 + 7
--> 4x + 0 = 20
--> 4x = 20
Then we divide both sides by 4, the coefficient of x. This leaves us with a coefficient of 1 for x, so we know we have solved for x.
We have 4x = 20
--> 4x/4 = 20/4
--> 1x = 5
--> x = 5
Example 4. Solve for x in the equation 3x + 6 = 5x - 2.
We can do the same thing in the previous example: isolate x by adding or subtracting terms from both sides of the equation. Let's first subtract 5x from both sides.
We have 3x + 6 = 5x - 2
--> 3x + 6 - 5x = 5x - 2 - 5x
--> -2x + 6 = 2
Next, we subtract 6 from both sides of the equation. This will isolate the -2x term.
We have -2x + 6 = 2
--> -2x + 6 - 6 = 2 - 6
--> -2x = -4
Now we divide both sides of the equation by -2. This gives us the answer for x.
We have -2x = -4
--> -2x/-2 = -4/-2
--> x = 2
Example 5. Eleanor bought 5 pints of vanilla ice cream and a fruit cake for a total of $45. If the fruit cake cost $25, how much did each pint of ice cream cost?
We can set the cost of one pint of ice cream to be x. Therefore, the cost of all five pints of ice cream is 5x. Adding 5x and the cost of the fruit cake is the total cost, $45.
We have 5x + 25 = 45
--> 5x + 25 - 25 = 45 - 25
--> 5x = 20
--> 5x/5 = 20/5
--> x = $4
Example 6. The bulk price of peaches at the farm is $2.50 per pound with a minimum purchase of 20 pounds. Margaret bought $80.00 of peaches. How many pounds more did Margaret's purchase exceed the minimum?
Let's first solve for the number of pounds of peaches, x, that Margaret bought using $80.00.
We have 2.50x = 80.00
--> 2.50x/2.50 = 80.00/2.50
--> x = 32 lbs of peaches
We now subtract 20 lbs from 32 lbs to get the number of pounds exceeding the minimum.
32 lbs - 20 lbs = 12 lbs over minimum
An algebraic inequality is a non-equal relation between two expressions. We manipulate terms similar to how we solve algebraic equations, but the answer gives us a range of values that satisfy the inequality.
Examples of algebraic inequalities include:
a > b
2x + 5 <= 14
ax^2 + (b-4)x + c >= 0
13m - 45n < 3p/4q
5abcd + e > 6mnp - qr
It is important to be aware that adding and subtracting a number on both sides of an inequality is performed the same way as we do in equations, but multiplying or dividing by a negative number requires us to flip the sign of the inequality.
Here is what happens when we multiply both sides of an inequality by a negative number.
Here is an example of how flipping signs work. This applies for multiplication as well.
Example 7. Aiden wants to save at least $600 for his new bike. He currently has $125. If he saves $15 every week, how many weeks does Aiden have to wait until he can afford his bike?
We can set x as the number of weeks Aiden has to wait. The amount of money he saves from all the weeks would be 15x. Adding $125 to 15x will give us at least 600, so we used ">=" to denote "at least".
We have 15x + 125 >= 600
--> 15x + 125 - 125 >= 600 - 125
--> 15x >= 475
--> 15x/15 >= 475/15
--> x >= 31 2/3 weeks
Example 8. Brandon has $500 in his bank account. Every week, he withdraws $40 for expenses. How many weeks can Brandon continue to draw money if he wants to maintain at least $200 in his account?
The total amount of money Brandon withdraws can be represented by 40x, with x being the number of weeks he withdraws his money. Since he is taking away money, we must subtract 40x from $500 dollars to get at least $200.
We have 500 - 40x >= 200
--> 500 - 40x - 500 >= 200 - 500
--> -40x >= -300
--> -40x/-40 >= -300/-40
--> x <= 7.5 weeks ------> notice that we flipped the sign!