Lesson 2: Expressions & Linear Equations/Inequalities

Expressions & Linear Equations/Inequalities

Expressions

Expressions are statements of math without equal signs or inequalities. They can be simplified or evaluated by combining like terms (more on that later).

Parts of an Expression Include...

• The "terms" which are the parts separated by operations (5x and -3 in this case).

• The "variables" are letters for which we substitute numbers. They are called variables because their values can vary, or differ.

• The "coefficients" are numbers which are attached to variables. They are multiplied by the variable once a value is substituted in.

• The "constants" are numbers which lack variables. They are called constants because their values do not change.

NOTE

When writing an expression, we arrange the terms in descending alphabetical order for variables, with the constant at the end of the expression. If there are multiples of the same variable, we write the expression in descending exponent order, then descending alphabetical order, then the constant last.

(i.e. If writing the expression  $5x-7y+14x^2-3$ , then we write it in this order:  $14x^2+5x-7y-3$ )

Simplify Expressions by Combining Like Terms

"Like terms" include any terms with the same variable and exponential value.

(ex. "2x²" and "5x" are not like terms, whereas "2x³" and "7x³" are)

Example Problems

•  $25x+14-17-6x$  

•  $6\left(n-2\right)-8n+40$  

•  $3g+9g^2-12g^2+g$  

Multiple Select

 $91x^3+22y^5-8x^2-45b^0-99x^3-70y$  
Which of the two terms above are like terms?

•  $22y^5$  

•  $-99x^3$  

•  $-8x^2$  

•  $-70y$  

•  $91x^3$  

Equations

Equations are two expressions set up with an equal sign between them.

To solve equations...

• you should add the opposites (in terms of their signs) of any terms which do not include the variable for which you're solving, then...

• multiply by the reciprocal of the coefficient of that variable.

• ("Reciprocal" means the inverse of a number. So, "2" is the reciprocal of "1/2" and "4/5" is the reciprocal of "5/4")

NOTE

Many times, before solving an equation, you will first need to simplify each expression on either side of the equal sign.

Also, if the variable you are solving for is on both sides, then add the opposite of one of its coefficients to each side in order to cancel it out on one side, then solve (as shown on next slide).

Example Problems

•  $24x+16=12$  

•  $4\left(q-5\right)=16$  

•  $7m+38=-5m-16$  

Inequalities

Inequalities are just like equations, except they use one of four inequality signs (>, <, >, <). The solutions are infinite and include every number above or below the one we find when we solve inequalities ("above or below" is determined by the inequality sign).

Reading Inequalities

Inequalities are read in a specific way. When written x > 2, we read it as, "x is greater than 2." When written x < 2, we read it as, "x is greater than or equal to 2."

Graphing Inequalities

The graphs of inequalities extend infinitely in one direction. It often helps to read the inequality out loud to determine the direction of the arrow for the graph of an inequality. Another trick is to use the direction of the inequality sign itself.

Graphing Inequalities

If the variable is on the left side of the inequality sign, the arrow points in the direction the graph arrow should go (as shown). If the variable is on the right side, then the graph arrow should go the opposite way.

Graphing Inequalities

Notice the circles within each graph. In the inequalities which include the phrase "equals," the circles are closed. In the inequalities which only state, "greater than" or "less than," the circles are open. This is because those second types of graphs do not include the number found, but everything right up until that number.

Multiple Choice

Which graph best describes x > 2?

(1 through 4 from top to bottom)

• 1

• 2

• 3

• 4

Solving Inequalities

Solving inequalities follows the same steps as solving equations:

1. Add the opposites of any terms which do not include the variable for which you're solving.

2. Multiply by the reciprocal of any coefficients of that variable.

However, there is one rule which changes...

Solving Inequalities

If you multiply (or divide) by a negative number, then the inequality sign flips!

(">" becomes "<" or "<" becomes ">")

Proof of the Sign Flip

Without the flip, this statement would be false instead of true.

Compound Inequalities

There are two kinds of compound inequalities, one is a conjunction ("and") and the other is a disjunction ("or"). For conjunctions, you solve all three sides at once and graph the solutions on the same number line. For a disjunction, you solve each inequality and graph the solutions on the same number line.

"And" Compound Inequality Graphs

For "And" compound inequality graphs, you place the circles on the indicated values and then connect them.

"Or" Compound Inequality Graphs

The arrows move away from each other with each circle on the indicated value.

Example Problems

•  $2x-3<-1$  

•  $15x+8>9x-22$  

•  $-8\le3x+1\le10$  

Multiple Choice

Solve the following inequality:
 $\frac{2}{3}x+9\le11$  

•  $x\le3$  

•  $x\ge\frac{4}{3}$  

•  $x\le\frac{4}{3}$  

•  $x\ge-3$  

Absolute Value Equations

Absolute Value Equations will almost always have two solutions. In order to solve an absolute value equation, you take the expression within the absolute value symbol and set it equal to both the positive constant on the other side of the equal sign and the negative of that same constant (as shown to the right). Then, you solve each equation to find each solution.

Solving Absolute Value Equations

Sometimes, you will need to first isolate the absolute value expression before you can solve for the variable. Do this before splitting the equation into two different equations (as shown to the right).

NOTE

If the constant on the opposite side of the equal sign is negative once the absolute value is isolated, then the equation has no solution!

(ex.  $\left|2x-9\right|=-11$  has no solution since 11 is negative)

Absolute Value Inequalities

When you solve an absolute value inequality, everything is the same as an absolute value equation except that you flip the inequality sign for the negative solution once the inequality is split in two.

Multiple Select

Solve for x (select both solutions)
 $-3\left|x-4\right|\ge-9$ 

•  $x\le7$  

•  $x\ge7$  

•  $x\le1$  

•  $x\ge1$