2. Equations and Inequalities

Find Intersections and Unions of Intervals

Linear Inequalities and Absolute Value Inequalities

Finding the union and intersections of intervals examples

Solve each inequality. Give the solution set in interval notation. . | 3x - 4 | ≥ 2

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

1 - x/2 > 4

Solve each inequality. Give the solution set in interval notation. | 5/3 - (1/2) x | > 2/9

Solve each inequality. Give the solution set in interval notation. See Example 4. 1≤(4x-5)/2<9

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

(- ∞, 5.5)

Match the inequality in each exercise in Column I with its equiva-lent interval notation in Column II. -2<x≤6

In Exercises 59–94, solve each absolute value inequality.

- 2|x - 4| ≥ - 4

Solve each equation or inequality. | 1.5x - 14| < 0

Solve each equation or inequality. |6 - 2x | + 1 = 3

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. 3(x+5)+1≥5+3x

The temperatures on the surface of Mars in degrees Celsius
approximately satisfy the inequality |C + 84 | ≤ 6. What range of temperatures
corresponds to this inequality?

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (1/3)x+(2/5)x-(1/2)(x+3)≤1/10

Solve inequalities with negative coefficients: dividing by negatives

In Exercises 59–94, solve each absolute value inequality.

|3x - 8| > 7

Solve each equation or inequality. | 4x+ 3| > 0

Solving inequalities with no solution or all real numbers as solutions

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

4(3x - 2) - 3x < 3(1 + 3x) - 7

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y = |2x - 5| + 1 and y > 9

Write each statement using an absolute value equation or inequality. . m is no more than 2 units from 7.

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (4x+7)/-3≤2x+5

In Exercises 59–94, solve each absolute value inequality.

5 > |4 - x|

Solve each inequality. Give the solution set in interval notation. | 3x - 4 | < 2

❖ Solving Linear Inequalities Made Easy ❖

In Exercises 59–94, solve each absolute value inequality.

|3 - (2/3)x| > 5

In Exercises 59–94, solve each absolute value inequality.

12 < |- 2x + 6/7| + 3/7

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

(x - 4)/6 ≥ (x - 2)/9 + 5/18

In Exercises 59–94, solve each absolute value inequality.

|(2x + 2)/4| ≥ 2

Using Interval Notation to Express Inequalities - Example 1

Interval Notation - A basic question!

Write each statement using an absolute value equation or inequality. q is no more than 8 units from 22.

Solve each equation or inequality. | 5x + 1/2 | -2 < 5

In Exercises 59–94, solve each absolute value inequality.

|2(x - 1) + 4| ≤ 8

In Exercises 59–94, solve each absolute value inequality.

- 4|1 - x| < - 16

In Exercises 15–26, use graphs to find each set.

(- ∞, 5) ⋃ [1, 8)

Match the inequality in each exercise in Column I with its equiva-lent interval notation in Column II. x<-6

An Introduction To Solving Linear Inequalities - Example 1

Solving Word Problems Involving Inequalities - Example 3

Solve each equation or inequality. |18- 3x | 4 < -13

Solve each equation or inequality. | 8x + 5| = 0

Solve each inequality. Give the solution set in interval notation. | 5 - 3x | > 7

Solve and graph the solution set on a number line: 3|2x-1| ≥ 21

Solve each inequality. Give the solution set using interval notation. -5x - 4≥3(2x-5)

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

(2, ∞)

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

2x - 11 < - 3(x + 2)

Solving Absolute Value Inequalities - Example 1

Explain why the equation | x | = √x² has infinitely many solutions.

Solve each equation or inequality. | 7 + 2x| = 0

Match the inequality in each exercise in Column I with its equiva-lent interval notation in Column II. x≤6

In Exercises 59–94, solve each absolute value inequality.

5|2x + 1| - 3 ≥ 9

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

8x - 11 ≤ 3x - 13

Solve each equation or inequality. | 2x+ 1/3 | +1 < 4

Solve each equation or inequality. | 10 - 4x | + 1 ≥ 5

Union and intersection of intervals

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y1 = (2/3)(6x - 9) + 4, y2 = 5x + 1, and y1 > y2

Solve each inequality. Give the solution set in interval notation. See Example 4. -4≤(x+1)/2≤5

Solve compound conjunction inequalities with 1 variable

Solve each equation or inequality. | 12- 5x | + 3 ≥ 9

In Exercises 15–26, use graphs to find each set.

(- 3, 0) ∩ [- 1, 2]

Solve each equation or inequality. | 12- 9x | ≥ -12

In Exercises 59–94, solve each absolute value inequality.

|2x - 6| < 8

An Introduction To Solving Linear Inequalities - Example 2

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. 8x-3x+2<2(x+7)

Solve and graph the solution set on a number line: (2x−3)/4 ≥ 3x/4 + 1/2

Using Interval Notation to Express Inequalities - Example 2

Writing Compound Inequalities Using Interval Notation - Example 2

Definition and Technique For Solving Linear Inequalities

Solve No Solution and All Real Number Inequalities

Solve each equation or inequality. |4x-12| ≥ -3

In Exercises 59–94, solve each absolute value inequality.

3|x - 1| + 2 ≥ 8

Drawing a Number Line to Match an Inequality

Solving Inequalities with No Solution or Infinite Solutions

Solve each equation or inequality. |8-5x| ≥ 2

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. 2-4x+5(x-1)<-6(x-2)

Solve each equation or inequality. | 4.3x + 9.8| < 0

Write each statement using an absolute value equation or inequality. r is no less than 1 unit from 29.

Interval Notation

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (2x-5)/-8≤1-x

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

[- 3, 1]

Inequalities with No Solution or All Real Numbers

Solve each inequality. Give the solution set in interval notation. | 7 - 3x | > 4

Solve each equation or inequality. | 10- 4x | ≥ 4

Solve each inequality. Give the solution set in interval notation. | 0.01x + 1 | < 0.01

When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.

Solve each equation or inequality. |4x + 3| - 2 = -1

Solving Absolute Value Inequalities, MORE Examples - Example 2

Solve each inequality. Give the solution set in interval notation. | (2/3)x + 1/2 | ≤ 1/6

Solving A Compound Linear Inequality

In Exercises 59–94, solve each absolute value inequality.

|x| > 3

Solve each equation or inequality. | 5x + 2 | - 2 < 3

Write each statement using an absolute value equation or inequality. p is at least 3 units from 1.

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | 3x^2 + x | = 14

Solve each equation or inequality. |7x+8| - 6 > -3

Match the inequality in each exercise in Column I with its equiva-lent interval notation in Column II . x≥-6

In Exercises 59–94, solve each absolute value inequality.

|3x + 5| < 17

In Exercises 15–26, use graphs to find each set.

[3, ∞) ⋃ (6, ∞)

When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.

Solve each equation or inequality. | 3x + 1 | - 1 < 2

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | x^4 + 2x^2 + 1 | < 0

Solve inequalities with 1 variable: graphing on a number line

Match the inequality in each exercise in Column I with its equiva-lent interval notation in Column II. 6≤x

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

3x - 7 ≥ 13

In Exercises 59–94, solve each absolute value inequality.

|x| < 3

To see how to solve an equation that involves the absolute value of a quadratic polynomial, such as | x^2 - x | = 6, work Exercises 83–86 in order. For x^2 - x to have an absolute value equal to 6, what are the two possible values that x may assume? (Hint: One is positive and the other is negative.)

In Exercises 59–94, solve each absolute value inequality.

4 + |3 - x/3| ≥ 9

In Exercises 59–94, solve each absolute value inequality.

|x + 3| ≤ 4

In Exercises 15–26, use graphs to find each set.

(- 3, 0) ⋃ [- 1, 2]

In Exercises 51–58, solve each compound inequality.

6 < x + 3 < 8

In Exercises 15–26, use graphs to find each set.

[3, ∞) ∩ (6, ∞)

Solve each inequality. Give the solution set in interval notation. See Example 4. 10≤2x+4≤16

Intersections & Unions of Intervals

Absolute Value Inequalities - How to Solve

In Exercises 59–94, solve each absolute value inequality.

|3(x - 1)/4| < 6

Solve each equation or inequality. | 3x- 7 | + 1 < -2

Solve each inequality. Give the solution set using interval notation. 11x≥2(x-4)

In Exercises 59–94, solve each absolute value inequality.

|x - 1| ≥ 2

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y1 = x/2 + 3, y2 = x/3 + 5/2, and y1 ≤ y2

Solve absolute value disjunction inequalities

Linear Inequalities

Solving Two - Step Linear Inequalities

In Exercises 103–104, use the graph of y = |4 - x| to solve each inequality. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/9627cb56ed6d8179-1676803885222.jpg" alt="" /></p>

|4 - x| ≥ 5

Match the inequality in each exercise in Column I with its equiva-lent interval notation in Column II. x^2≥0

In Exercises 59–94, solve each absolute value inequality.

3 ≤ |2x - 1|

Solve each inequality. Give the solution set using interval notation. 7x-2(x-3) ≤5(2-x)

Solve each inequality. Give the solution set in interval notation. | 3/5 + x | < 1

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

(- ∞, 3)

Solve each equation or inequality. |-5x + 7 | - 4 < -6

Solving Word Problems Involving Inequalities - Example 2

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

[- 3, ∞)

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

1 - (x + 3) ≥ 4 - 2x

Solve absolute value conjunction inequalities

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

[- 5, 2)

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

x/4 - 3/2 ≤ x/2 + 1

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y = 8 - |5x + 3| and y is at least 6

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

5x + 11 < 26

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. -(2/3)x-(1/6)x+(2/3)(x+1)≤4/3

Solve each inequality. Give the solution set in interval notation. 5| x + 1 | > 10

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.

(1, 6]

Solve each inequality. Give the solution set in interval notation. | 1/2 - x | ≤ 2

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y = 2x - 11 + 3(x + 2) and y is at most 0

In Exercises 51–58, solve each compound inequality.

- 3 ≤ x - 2 < 1

Solving 3 Basic Linear Inequalities

Solve each inequality. Give the solution set using interval notation. -8 >3x-5>-12

Solve compound disjunction inequalities with 1 variable

Solve each inequality. Give the solution set in interval notation. See Example 4. 2>-6x+3>-3

In Exercises 59–94, solve each absolute value inequality.

|x - 1| ≤ 2

Solving Algebraic Inequalities

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

4(x + 1) + 2 ≥ 3x + 6

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

-9x ≥ 36

Solve each equation or inequality. |7x-3| > 4

Solve each equation or inequality. | 4 - 4x | + 2 = 4

Solve each equation or inequality. | 6- 3x | 4 < -11

Solve each equation or inequality. |8 - 3x| - 3 = -2

An Introduction To Solving Linear Inequalities - Example 3

In Exercises 51–58, solve each compound inequality.

- 3 ≤ (2/3)x - 5 < - 1

Solving Basic Linear Inequalities Involving Multiplication or Division

Solve each inequality. Give the solution set using interval notation. 5 ≤ 2x -3 ≤ 7

In Exercises 15–26, use graphs to find each set.

(- ∞, 5) ∩ [1, 8)

In Exercises 51–58, solve each compound inequality.

- 11 < 2x - 1 ≤ - 5

In Exercises 59–94, solve each absolute value inequality.

1 < |2 - 3x|

Solve each inequality. Give the solution set in interval notation. | 7 - 3x | ≤ 4

Solve each inequality. Give the solution set in interval notation. | 5 - 3x | ≤ 7

Solving Linear Absolute Value Equations and Inequalities

Solve each inequality. Give the solution set in interval notation. See Example 4. -5<5+2x<11

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