Problem 59
Question
Determine whether each value of \(x\) is a solution of the inequality. \(5(x-2)+1<12\) (a) \(x=4\) (b) \(x=1\) (c) \(x=5\) (d) \(x=-3\)
Step-by-Step Solution
Verified Answer
Out of the provided values of x, only \(x = 4\), \(x = 1\), and \(x = -3\) are solutions to the inequality \(5(x - 2) + 1 < 12\).
1Step 1: Substitute \(x=4\) in the inequality
Replace \(x\) with \(4\) and simplify the expression as follows: \(5(4 - 2) + 1 < 12 \)10 + 1 < 12 \Given that 11 is less than 12, \(x = 4\) is a solution to the inequality.
2Step 2: Substitute \(x=1\) in the inequality
Replace \(x\) with \(1\) in the inequality: \(5(1 - 2) + 1 < 12 \)-5 + 1 < 12 \Given that \(-4\) is less than \(12\), \(x = 1\) is a solution to the inequality.
3Step 3: Substitute \(x=5\) in the inequality
Replace \(x\) with \(5\) in the inequality: \(5(5 - 2) + 1 < 12 \15 + 1 < 12 \Given that \(16\) is not less than \(12\), \(x = 5\) is not a solution to the inequality.
4Step 4: Substitute \(x=-3\) in the inequality
Replace \(x\) with \(-3\) in the inequality: \(5(-3 - 2) + 1 < 12 \-25 + 1 < 12 \Given that \(-24\) is less than \(12\), \(x = -3\) is a solution to the inequality.
Key Concepts
Inequality Solution CheckingSubstitution MethodLinear InequalitiesAlgebraic Expressions
Inequality Solution Checking
Understanding how to verify solutions to an inequality is crucial in algebra. When given an inequality, such as \(5(x-2)+1<12\), we need to determine if a specific value of \(x\) makes the inequality true. This process is known as inequality solution checking. To do this, you substitute the given values of \(x\) one by one into the inequality and simplify.
For example, if you substitute \(x=4\) into the inequality and simplify to find that \(11<12\), this means that when \(x\) is \(4\), the inequality holds true, and thus \(4\) is a solution. It's always important to check all provided values, because while some may satisfy the inequality, others may not. Remember, determining the correctness of a solution is a matter of comparing the simplified expressions and ensuring the inequality relationship remains true.
For example, if you substitute \(x=4\) into the inequality and simplify to find that \(11<12\), this means that when \(x\) is \(4\), the inequality holds true, and thus \(4\) is a solution. It's always important to check all provided values, because while some may satisfy the inequality, others may not. Remember, determining the correctness of a solution is a matter of comparing the simplified expressions and ensuring the inequality relationship remains true.
Substitution Method
The substitution method is a fundamental technique in algebra, especially for solving equations and inequalities. It involves replacing a variable with a given number to determine if the equation or inequality is true for that number. The steps are straightforward: take the given value of the variable, replace every instance of the variable in the equation or inequality with that value, and then perform the arithmetic to simplify.
For instance, with the inequality \(5(x-2)+1<12\), if you wish to test whether \(x=1\) is a solution, you would substitute \(1\) for every \(x\) in the inequality, leading to \(-5+1<12\), which simplifies to \(-4<12\). Thus, \(x=1\) is indeed a solution. Consistent application of this method helps in finding which values satisfy the inequality.
For instance, with the inequality \(5(x-2)+1<12\), if you wish to test whether \(x=1\) is a solution, you would substitute \(1\) for every \(x\) in the inequality, leading to \(-5+1<12\), which simplifies to \(-4<12\). Thus, \(x=1\) is indeed a solution. Consistent application of this method helps in finding which values satisfy the inequality.
Linear Inequalities
Linear inequalities are similar to linear equations but instead of an equal sign, they have an inequality symbol (such as <, >, \(\leq\), or \(\geq\)). They represent a range of values that satisfy the conditions of the inequality, rather than a single solution, which is the case with an equation.
The inequality \(5(x-2)+1<12\) is a linear inequality because it can be simplified to a form similar to \(ax+b
The inequality \(5(x-2)+1<12\) is a linear inequality because it can be simplified to a form similar to \(ax+b
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operators (such as +, -, *, and /) that represent a specific quantity. Unlike an equation, an expression doesn't have an equality sign and thus doesn't assert a claim that can be true or false.
If we consider the inequality \(5(x-2)+1<12\), the portion \(5(x-2)+1\) on its own is an example of an algebraic expression. It consists of the variable \(x\), numbers, and the operations of multiplication and addition. In the context of inequalities, algebraic expressions result in a range of values rather than a single outcome. Grasping the concept of algebraic expressions is key to mastering algebra as they form the foundation of equations and inequalities.
If we consider the inequality \(5(x-2)+1<12\), the portion \(5(x-2)+1\) on its own is an example of an algebraic expression. It consists of the variable \(x\), numbers, and the operations of multiplication and addition. In the context of inequalities, algebraic expressions result in a range of values rather than a single outcome. Grasping the concept of algebraic expressions is key to mastering algebra as they form the foundation of equations and inequalities.
Other exercises in this chapter
Problem 58
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