Linear Inequalities in One Variable: Videos & Practice Problems

Introduction to Linear Inequalities

Linear inequalities are closely related to linear equations but differ by using inequality symbols instead of an equal sign. While a linear equation typically takes the form \(ax + b = c\), a linear inequality replaces the equal sign with symbols such as \(>\), \(<\), \(\geq\), or \(\leq\), resulting in expressions like \(ax + b > c\). This subtle change transforms the problem from finding a single solution to identifying a range of solutions.

In a linear equation, solving for \(x\) yields a specific value that satisfies the equation. For example, solving \(2x - 6 = 0\) gives \(x = 3\). However, when dealing with a linear inequality such as \(2x - 6 > 0\), the solution is not a single number but all values of \(x\) that make the inequality true. In this case, the solution is \(x > 3\), meaning any number greater than 3 satisfies the inequality.

To verify solutions of linear inequalities, you can substitute values from the solution range back into the inequality. For instance, plugging in \(x = 5\) into \(2x - 6 > 0\) results in \(2(5) - 6 = 4\), and since \(4 > 0\) is true, \(x = 5\) is indeed a valid solution. This demonstrates that the solution set of a linear inequality is a continuous range rather than a discrete point.

Understanding linear inequalities involves recognizing their form and interpreting their solutions as ranges of values. This concept builds on the foundation of linear equations, extending problem-solving skills to include inequalities. Mastery of these ideas is essential for graphing inequalities, solving real-world problems, and progressing to more advanced algebraic concepts.

Representing Solutions to Linear Inequalities

When solving inequalities, the solution is not a single value but a range of values. Understanding how to represent and visualize these solution sets is essential. One common way to express inequalities is through set builder notation, which uses curly brackets to define the set of all values that satisfy the inequality. For example, the inequality x > 3 is written as { x | x > 3 }, which reads as "the set of x such that x is greater than 3." The vertical bar means "such that."

Another way to visualize inequalities is on a number line. To represent x > 3, you start at 3 and draw an arrow pointing to the right, indicating all values greater than 3. Since 3 itself is not included, a parenthesis is used at 3 to show exclusion. This use of parentheses to exclude endpoints is consistent with interval notation, where the solution is written as (3, ∞). Here, parentheses around 3 indicate it is not included, and parentheses around infinity are always used because infinity is not a specific number and cannot be included.

For inequalities that include the boundary value, such as x ≥ 3, set builder notation remains similar: { x | x ≥ 3 }. On the number line, the point at 3 is included, so a square bracket or a closed circle is used to indicate inclusion. In interval notation, this is written as [3, ∞), where the square bracket at 3 shows inclusion, and the parenthesis at infinity remains.

When dealing with inequalities where x is less than a value, such as x < 3, the direction of the arrow on the number line points left, toward smaller values. The notation follows the same rules: parentheses indicate exclusion, so the interval notation is (−∞, 3). For x ≤ 3, the interval notation becomes (−∞, 3], with a square bracket indicating that 3 is included.

It is important to remember that in interval notation, infinity and negative infinity always use parentheses because they are not actual numbers and cannot be included in the set. On number lines, open circles represent excluded endpoints, while closed circles represent included endpoints. Although parentheses and square brackets are more commonly used in graphs and interval notation, recognizing open and closed circles is helpful for interpreting inequalities visually.

Mastering these notations—set builder, number line, and interval notation—allows for clear communication and understanding of solution sets for linear inequalities, which is foundational for solving and graphing inequalities effectively.

Which inequality matches the graph?

• A. $x\(\ge\)1$
• B. $x\(\le\)1$
• C. $x>1$
• D. $x<1$

Answer: (A)

Which inequality matches the graph?

• A. $x\(\ge\)1$
• B. $x\(\le\)1$
• C. $x>1$
• D. $x<1$

Answer: (A)

Use the number line to graph the following inequality.

$x\ge 6$

• A.
• B.
• C.
• D.

Answer: (C)

Use the number line to graph the following inequality.

$x\ge -7$

• A.
• B.
• C.
• D.

Answer: (A)

Write the following in interval notation.
$x\ge 0$

• A. $\(\left\)(-\(\infty\),0\(\right\))$
• B. $\(\left\[\lbrack\)0,\(\infty\]\right\))$
• C. $\(\left\)(0,\(\infty\]\right\))$
• D. $\(\left\)(-\(\infty\),0\(\right\]\rbrack\)$

Answer: (B)

Write the following in interval notation.

• A. $\(\left\)(-\(\infty\),7\(\right\]\rbrack\)$
• B. $\(\left\[\lbrack\)7,\(\infty\]\right\))$
• C. $\(\left\)(7,\(\infty\]\right\))$
• D. $\(\left\)(-\(\infty\),7\(\right\))$

Answer: (D)

Rewrite the following as an inequality statement.
$(-\infty ,2]$

• A. $x\(\le\)2$
• B. $x<2$
• C. $x\(\ge\)2$
• D. $x>2$

Answer: (A)

Rewrite the following as an inequality statement.
$(1,\infty )$

• A. $x\(\ge\)1$
• B. $x\(\le\)1$
• C. $x>1$
• D. $x<1$

Answer: (C)

Introduction to Linear Inequalities Example 1

Understanding how to translate between inequalities, interval notation, and graphical representations is essential for mastering the concept of inequalities. When given an inequality such as x > -2, it corresponds to the interval notation (-2, ∞), where the parenthesis indicates that -2 is not included in the set. This matches a graph starting just after -2 and extending infinitely to the right.

For inequalities that include the boundary value, such as x ≥ -5, the interval notation uses a square bracket to denote inclusion: [-5, ∞). This means the set includes -5 and all values greater than -5, which is reflected in the graph by a solid point at -5 and an arrow extending to infinity.

Intervals like (4, ∞) represent values greater than 4 but not including 4 itself, indicated by a parenthesis. The corresponding graph starts just after 4 and extends to the right, without including the point at 4.

When dealing with values less than a certain number, such as values less than -2, the interval notation is (-∞, -2), with a parenthesis at -2 to show it is excluded. The graph will have an arrow pointing left from -2, starting just after -2 and extending towards negative infinity.

Intervals that include the endpoint, like (-∞, 3], include all values less than or equal to 3. The square bracket at 3 indicates inclusion, and the graph reflects this with a solid point at 3 and an arrow pointing left towards negative infinity.

It is important to recognize that inequalities can be written in different forms but represent the same set. For example, -1 ≤ x is equivalent to x ≥ -1. The interval notation for this is [-1, ∞), and the graph shows a solid point at -1 with an arrow extending to the right.

By understanding the relationship between inequalities, interval notation, and graphs, one can confidently interpret and convert between these forms. This skill is fundamental in algebra and helps in solving and visualizing inequality problems effectively.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable using properties similar to those applied in linear equations, but with attention to the inequality symbols. The addition and subtraction properties of inequalities state that if A is greater than B, then adding or subtracting the same value C to both sides preserves the inequality: \(A + C > B + C\) and \(A - C > B - C\). For example, to solve \(x - 3 > 11\), add 3 to both sides to isolate x, resulting in \(x > 14\).

Multiplication and division properties also apply, where multiplying or dividing both sides of an inequality by a positive number maintains the inequality's direction. For instance, if \(2 < \frac{x}{5}\), multiplying both sides by 5 yields \(10 < x\). However, a critical distinction arises when multiplying or dividing by a negative number: the inequality symbol must be reversed to maintain a true statement. This is because multiplying or dividing by a negative value changes the sign of each side, necessitating flipping the inequality symbol. For example, solving \(-7x \geq 21\) involves dividing both sides by \(-7\), which flips the inequality to \(x \leq -3\).

Understanding these properties allows for effective solving of linear inequalities by isolating the variable while carefully managing the inequality direction, especially when negative multiplication or division is involved. This approach ensures accurate solutions and reinforces the foundational connection between solving linear equations and inequalities.

Solving Linear Inequalities Example 2

To solve the linear inequality x + 8 ≤ 12 - x, start by gathering all variable terms on one side and constants on the other. Add x to both sides to eliminate the negative x on the right, resulting in 2x + 8 ≤ 12. Next, subtract 8 from both sides to isolate the variable term: 2x ≤ 4. Finally, divide both sides by 2 to solve for x, giving x ≤ 2.

Graphing this solution on a number line involves marking the point at 2 and shading all values less than or equal to 2. Since the inequality includes equality, use a closed circle or square bracket at 2 to indicate that 2 is part of the solution set.

Expressed in interval notation, the solution is (-∞, 2]. Parentheses are used with infinity because infinity is not a number and cannot be included, while the square bracket at 2 shows that 2 is included in the interval.

This process demonstrates how to solve linear inequalities by applying inverse operations, graph the solution set accurately, and express the solution in interval notation, which is essential for understanding ranges of values that satisfy inequalities.

Solve the following linear inequalities using the addition and subtraction properties of equality.

• A. $x<2$
• B. $x>2$
• C. $x<1.8$
• D. $x\(\le\)1.8$

Answer: (A)

Solve the following linear inequalities and write the solution in interval notation.
$2(x+4)\le 3(x-1)+x$

• A. $\(\frac{11}{2}\]\ge\) x$
• B. $\(\frac{11}{2}\]\le\) x$
• C. $\(\frac{11}{4}\]\le\) x$
• D. $\(\frac{11}{4}\]\ge\) x$

Answer: (B)

Solve the following linear inequalities and write the solution in interval notation.

• A. $x<-\(\frac\)52$
• B. $x<-\(\frac{18}{5}\)$
• C. $x>-\(\frac{18}{5}\)$
• D. $x>-\(\frac\)52$

Answer: (C)

Solving Linear Inequalities Example 3

When solving inequalities, it is important to isolate the variable terms on one side and the constants on the other. Consider the inequality \(2x - 5 < 2x + 10\). By subtracting \$2x\( from both sides, the variable terms cancel out, leaving \)-5 < 10\(. Since this statement is always true, the inequality holds for all real numbers. This means the solution set includes every real number because no matter what value of \)x\( is substituted, the inequality remains valid.

In contrast, examine the inequality \(x + 2 \geq x + 8\). Subtracting \)x\( from both sides eliminates the variable terms, resulting in \(2 \geq 8\). This is a false statement, indicating that the inequality has no solution. Such inequalities are called inconsistent because no value of \)x$ can satisfy them.

These examples illustrate key concepts in solving linear inequalities. When variables cancel and the resulting statement is true, the solution is all real numbers, reflecting an identity inequality. When variables cancel and the resulting statement is false, the inequality is inconsistent, meaning no solution exists. Understanding these outcomes helps in analyzing inequalities effectively and recognizing when solutions are universal or nonexistent.

Solving Three Part Inequalities

Solving three-part inequalities involves working with an inequality where the variable expression is situated between two inequality symbols, creating three distinct parts: the left side, the center, and the right side. For example, in the inequality −14 ≤ 2x − 10 < 2, the expression 2x − 10 is between two inequality symbols, forming a three-part inequality.

The key to solving these inequalities is to isolate the variable in the center, aiming for a solution that looks like a ≤ x ≤ b or a < x < b, where a and b are numbers. The process mirrors solving simple inequalities but with an important adjustment: any operation performed on one part of the inequality must be applied to all three parts to maintain balance.

For instance, to solve −14 ≤ 2x − 10 < 2, start by eliminating the constant term from the center. Adding 10 to all three parts yields −14 + 10 ≤ 2x − 10 + 10 < 2 + 10, simplifying to −4 ≤ 2x < 12. Next, to isolate x, divide all three parts by 2, resulting in −2 ≤ x < 6. This final inequality shows the variable x isolated between two numbers.

The solution can be expressed in set-builder notation as { x | −2 ≤ x < 6 }, which reads as "the set of all x such that x is greater than or equal to −2 and less than 6." In interval notation, this is written as [−2, 6), where the square bracket [ indicates that −2 is included in the solution set (because of the "less than or equal to" symbol), and the parenthesis ) indicates that 6 is not included (due to the strict "less than" symbol).

Graphically, this solution is represented on a number line with a closed bracket or filled circle at −2 and an open parenthesis or open circle at 6, connected by a line segment between these two points. Unlike inequalities involving infinity, there are no arrows extending beyond these points since the solution is bounded.

Understanding three-part inequalities enhances problem-solving skills by reinforcing the importance of performing equivalent operations across all parts of the inequality. This ensures the inequality remains balanced and the solution accurate. Mastery of this concept allows for confident manipulation and interpretation of compound inequalities, which frequently appear in algebra and real-world applications.

Solve the following inequalities and express the answer in interval notation.
$-4\le 2x+1\le 7$

• A. $[-6,8]$
• B. $[-\frac{3}{2},4]$
• C. $[-\frac{5}{2},3]$
• D. $[-10,6]$

Answer: (C)

Solve the following inequalities and express the answer in interval notation.

• A. $[4,10)$
• B. $(4,10]$
• C. $[0,6)$
• D. $(0,6]$

Answer: (B)

Solve the following inequalities and graph the solution.

• A.
• B.
• C.
• D.

Answer: (A)

Solve the following inequalities and graph the solution.

• A.
• B.
• C.
• D.

Answer: (D)

Solving Three Part Inequalities Example 4

To solve the three-part inequality 4 < 6 − x ≤ 8, the objective is to isolate the variable x in the center of the inequality. This requires performing the same operation on all three parts of the inequality to maintain balance.

Starting with the inequality, subtract 6 from each part to eliminate the constant term next to x:

\[4 - 6 < 6 - x - 6 \leq 8 - 6\]

which simplifies to:

\[-2 < -x \leq 2\]

Next, to isolate x, divide all parts of the inequality by −1. Remember, dividing or multiplying an inequality by a negative number reverses the inequality signs:

\[\frac{-2}{-1} > \frac{-x}{-1} \geq \frac{2}{-1}\]

which simplifies to:

\[2 > x \geq -2\]

For clarity and conventional notation, rewrite the inequality by reading it from left to right:

\[-2 \leq x < 2\]

This means x is greater than or equal to −2 and less than 2.

Expressing this solution in interval notation, since x includes −2 but does not include 2, the interval is:

\[[-2, 2)\]

Here, the square bracket [ indicates that −2 is included in the solution set, while the parenthesis ) indicates that 2 is excluded.

Graphically, this solution is represented on a number line by a solid point or bracket at −2, a line segment extending to 2, and an open parenthesis or hollow point at 2, showing that 2 is not part of the solution.

Understanding how to manipulate inequalities, especially the rule that dividing or multiplying by a negative number reverses inequality signs, is crucial for solving compound inequalities and expressing their solutions accurately in interval notation and graphs.

Here's what students ask on this topic:

What is a linear inequality and how does it differ from a linear equation?

A linear inequality is similar to a linear equation but instead of an equal sign, it uses inequality symbols such as >, <, ≥, or ≤. For example, a linear equation looks like $\mathrm{2x}-6=0$, while a linear inequality might be $\mathrm{2x}-6>0$. The key difference is that the solution to a linear equation is a single value of $x$ that makes the equation true, whereas the solution to a linear inequality is a range of values that satisfy the inequality. For example, $x=3$ solves the equation, but for the inequality, the solution is $x>3$, meaning any value greater than 3 works. This range of solutions can be expressed in different notations and visualized on a number line.

How do you represent the solution of a linear inequality using interval notation?

Interval notation is a concise way to represent the solution set of a linear inequality. For example, if the solution is $x>3$, the interval notation is $(3,\; 2ext\{infinity\}2)$. Parentheses $(\; )$ indicate that the endpoint is not included, so 3 is excluded here. If the inequality includes the endpoint, such as $x\ge 3$, we use a square bracket $[\; ]$ to include it, so the interval is $[3,\; 2ext\{infinity\}2)$. Infinity always uses parentheses because it is not a finite number and cannot be included. Similarly, for inequalities like $x<3$, the interval is $(-2ext\{infinity\},\; 3)$, and if it includes 3, it is $(-2ext\{infinity\},\; 3]$. This notation helps clearly define the range of values that satisfy the inequality.

What is the rule for solving linear inequalities when multiplying or dividing by a negative number?

When solving linear inequalities, if you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality symbol. For example, if you have $-7x\ge 21$, dividing both sides by $-7$ requires flipping the inequality: $x2ext\{less\; than\; or\; equal\; to\}2-3$. This happens because multiplying or dividing by a negative number changes the order of the values. Forgetting to flip the inequality symbol leads to incorrect solutions. This rule is unique to inequalities and does not apply to equations. Always remember to flip the inequality sign when multiplying or dividing by a negative to maintain a true statement.

How do you solve a three-part linear inequality?

A three-part linear inequality has the variable expression between two inequality symbols, such as $-142ext\{less\; than\; or\; equal\; to\}2\; 2x\; -\; 10\; 2ext\{less\; than\}22$. To solve it, perform the same operation on all three parts to isolate the variable in the center. For example, add 10 to all three sides: $-14+\; 10\; 2ext\{less\; than\; or\; equal\; to\}2\; 2x\; 2ext\{less\; than\}22+\; 10$, simplifying to $-42ext\{less\; than\; or\; equal\; to\}2\; 2x\; 2ext\{less\; than\}212$. Then divide all sides by 2: $-22ext\{less\; than\; or\; equal\; to\}2\; x\; 2ext\{less\; than\}26$. The solution shows the range of values for $x$. This solution can be expressed in set builder notation, interval notation, or graphed on a number line. The key is to apply every operation to all three parts to maintain the inequality's balance.

What are the different notations used to represent solutions of linear inequalities?

Solutions to linear inequalities can be represented in three main notations: set builder notation, interval notation, and graphing on a number line. Set builder notation uses curly brackets and a vertical bar to describe the set, for example, $\{x\; |\; x\; >\; 3\}$, read as "the set of all x such that x is greater than 3." Interval notation uses parentheses and brackets to indicate ranges, such as $(3,\; 2ext\{infinity\}2)$ for values greater than 3, excluding 3, or $[3,\; 2ext\{infinity\}2)$ if 3 is included. Parentheses exclude endpoints, brackets include them. Graphing uses a number line with arrows showing the range and parentheses or brackets (or open/closed circles) to indicate whether endpoints are included or excluded. Understanding these notations helps visualize and communicate the solution sets effectively.