Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5.
x² + 2x - 8 = 0
Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.6.93
Solve each inequality. Give the solution set using interval notation. See Example 10. -5 < 5 + 2x < 11
Verified step by step guidance1
Start by understanding that the compound inequality \(-5 < 5 + 2x < 11\) means both inequalities \(-5 < 5 + 2x\) and \(5 + 2x < 11\) must be true simultaneously.
Isolate the variable \(x\) in the first inequality: subtract 5 from all parts to get \(-5 - 5 < 2x\), which simplifies to \(-10 < 2x\).
Next, isolate \(x\) in the second inequality: subtract 5 from all parts to get \(2x < 11 - 5\), which simplifies to \(2x < 6\).
Now, solve for \(x\) in both inequalities by dividing all parts by 2 (note that dividing by a positive number does not change the inequality direction): from \(-10 < 2x\) we get \(-5 < x\), and from \(2x < 6\) we get \(x < 3\).
Combine the two results to write the solution set as \(-5 < x < 3\), which in interval notation is expressed as \((-5, 3)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Compound Inequalities
A compound inequality involves two inequalities joined together, such as -5 < 5 + 2x < 11. To solve it, you treat it as two separate inequalities and find the values of the variable that satisfy both simultaneously.
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Finding the Domain and Range of a Graph
Solving Linear Inequalities
Solving linear inequalities involves isolating the variable on one side by performing inverse operations, similar to solving equations, but remembering to reverse the inequality sign when multiplying or dividing by a negative number.
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Interval Notation
Interval notation is a way to express the solution set of inequalities using intervals. It uses parentheses for values not included and brackets for values included, clearly showing the range of possible solutions.
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Related Practice
Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(x + 7) = 2(x + 12) + 2(x + 1)
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5.
9x² - 12x + 4 = 0
Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. 2x² - x - 28 = 0
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 5x +2 ≤ -48
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Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 2(x - 8) = 3x - 16