Algebra (Comprehensive Review)

Chapter 1

Algebra

Course: Sat Prep
Subject: Math
Chapter: Algebra

By the end of this lesson, you will be able to:

• Solve one-variable linear inequalities.
• Solve compound inequalities involving AND and OR.
• Express solution sets using interval notation.
• Graph inequalities correctly on a number line.
• Translate real-world situations into linear inequalities.
• Verify whether a value satisfies an inequality.

What Is a Linear Inequality?

A linear inequality is a mathematical statement that compares two algebraic expressions using an inequality symbol instead of an equal sign. Unlike a linear equation, which has one or more exact solutions, a linear inequality has a range of solutions. Every value in this range satisfies the inequality.

The highest exponent of the variable in a linear inequality is 1, making it linear.

The general forms of linear inequalities are

$ax+b>c$ $ax+b<c$ $ax+b\ge c$ $ax+b\le c$

where

• $a\neq0$
• $x$ is the variable
• $a$, $b$, and $c$ are constants.

For example,

$x>4$

means every number greater than 4 is a solution.

Some possible solutions are

$5,\;6,\;7,\;8,\;9,\;10,\ldots$

Unlike equations, which usually have one or more exact solutions, inequalities often have infinitely many solutions.

Inequality Symbols

| Symbol | Meaning | Example | | $>$ | Greater than | $x>5$ | | $<$ | Less than | $x<8$ | | $\ge$ | Greater than or equal to | $x\ge3$ | | $\le$ | Less than or equal to | $x\le9$ |

Linear Equations vs. Linear Inequalities

| Linear Equation | Linear Inequality | | Uses the equals sign (=) | Uses inequality symbols ($>,<,\ge,\le$) | | Usually has one or more exact solutions | Usually has infinitely many solutions | | Solution is a value or set of values | Solution is often a range of values | | Example: $2x+4=10$ | Example: $2x+4>10$ |

Solving One-Variable Linear Inequalities

The steps for solving a linear inequality are almost identical to those used when solving a linear equation.

You may:

• Add the same number to both sides.
• Subtract the same number from both sides.
• Multiply both sides by the same positive number.
• Divide both sides by the same positive number.

However, there is one very important rule.

Whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

This is one of the most frequently tested concepts on the Digital SAT.

Example 1

Solve the inequality

$4x-7>13$

Solution

Add 7 to both sides.

$4x>20$

Divide both sides by 4.

$x>5$

Therefore,

$\boxed{x>5}$

The graph of the solution is shown below.

The open circle indicates that 5 is not included in the solution set. The shaded arrow points to the right because the solution consists of numbers greater than 5.

Example 2

Solve the inequality

$-3x+9\ge0$

Solution

Subtract 9 from both sides.

$-3x\ge-9$

Divide both sides by -3.

Since you are dividing by a negative number, reverse the inequality sign.

$x\le3$

Therefore,

$\boxed{x\le3}$

The graph of the solution is shown below.

The closed circle indicates that 3 is included in the solution set.

Multiplying or Dividing by a Negative Number

The most important rule when solving inequalities is remembering to reverse the inequality symbol whenever both sides are multiplied or divided by a negative number.

Consider the statement

$5>2$

Multiply both sides by $-1$.

$-5<-2$

Notice that the direction of the inequality changes.

This happens because multiplying by a negative number reverses the order of all real numbers on the number line.

Example

Solve

$-4x>20$

Divide both sides by $-4$.

Reverse the inequality sign.

$x<-5$

This is the correct solution.

Remember

| Operation | Reverse the Inequality? | | Add a number | No | | Subtract a number | No | | Multiply by a positive number | No | | Divide by a positive number | No | | Multiply by a negative number | Yes | | Divide by a negative number | Yes |

Compound Inequalities

A compound inequality combines two inequalities into one mathematical statement.

There are two types of compound inequalities.

AND Inequalities

An AND inequality means both conditions must be true at the same time.

Example:

$2<x\le7$

This means the solution includes all numbers greater than 2 and less than or equal to 7.

Graphically,

Example

Solve

$5<2x+1\le13$

Subtract 1 from every part.

$4<2x\le12$

Divide every part by 2.

$2<x\le6$

The solution is

$\boxed{2<x\le6}$

OR Inequalities

An OR inequality means either condition can be true.

Example

$x<-3$

or

$x>5$

Graphically,

The solution contains two separate regions on the number line.

Interval Notation

Interval notation is another way to write solution sets.

Instead of writing inequalities, mathematicians often use intervals.

Rules for Interval Notation

• Parentheses ( ) mean the endpoint is not included.
• Brackets [ ] mean the endpoint is included.
• Infinity and negative infinity always use parentheses because infinity is not a real number.

Common Conversions

| Inequality | Interval Notation | | $x>4$ | $(4,\infty)$ | | $x\ge4$ | $[4,\infty)$ | | $x<3$ | $(-\infty,3)$ | | $x\le3$ | $(-\infty,3]$ | | $2<x\le6$ | $(2,6]$ | | $x<-4$ or $x>5$ | $(-\infty,-4)\cup(5,\infty)$ |

Example

Write the inequality

$x\ge8$

using interval notation.

Solution

Since 8 is included,

$\boxed{[8,\infty)}$

Graphing Linear Inequalities

Graphing inequalities makes it easier to visualize all possible solutions.

Rules for Graphing

• Use an open circle (○) for $<$ or $>$.
• Use a closed circle (●) for $\le$ or $\ge$.
• Shade to the left for values less than the endpoint.
• Shade to the right for values greater than the endpoint.

Examples

For

$x<4$

For

$x\ge-2$

Graphing is an important skill because many SAT questions present inequalities visually rather than algebraically.

Word Problems

Many SAT questions require translating words into mathematical inequalities.

Understanding common phrases will help you write the correct inequality.

| Phrase | Mathematical Symbol | | At least | $\ge$ | | At most | $\le$ | | More than | $>$ | | Less than | $<$ | | No more than | $\le$ | | No less than | $\ge$ | | Exceeds | $>$ |

Example 1

A concert hall can seat no more than 450 people.

Let $p$ represent the number of people.

Write an inequality.

Solution

"No more than" means less than or equal to.

$\boxed{p\le450}$

Example 2

A student must score at least 70 points to pass an examination.

Let $s$ represent the student's score.

Write an inequality.

Solution

"At least" means greater than or equal to.

$\boxed{s\ge70}$

Common Mistakes

One of the most common mistakes is forgetting to reverse the inequality sign after multiplying or dividing by a negative number.

Incorrect:

$-2x>10$ $x>-5$

Correct:

$x<-5$

Another common mistake is using a closed circle when graphing a strict inequality such as $x>4$. Since 4 is not included, the graph should use an open circle.

Students also frequently confuse AND and OR compound inequalities. An AND inequality represents the overlap between two conditions, while an OR inequality represents values that satisfy either condition.

Lesson Summary

In this lesson, you learned how to solve one-variable linear inequalities, apply the rule for reversing the inequality sign when multiplying or dividing by a negative number, solve compound inequalities using AND and OR, express solutions in interval notation, graph inequalities on a number line, and model real-world situations using inequalities.