Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ -3<2 x-5<7 $$

Step-by-Step Solution

Answer

The solution is \((1, 6)\).

Step 1: Analyze the Compound Inequality

The given compound inequality is:\[-3 < 2x - 5 < 7\]This means we need to solve the two inequalities simultaneously:1. \(-3 < 2x - 5\)2. \(2x - 5 < 7\).

Step 2: Solve the First Inequality

Start with the inequality \(-3 < 2x - 5\).Add 5 to both sides to isolate the term with \(x\):\[ -3 + 5 < 2x \]This simplifies to:\[ 2 < 2x \]Next, divide both sides by 2 to solve for \(x\):\[ 1 < x \] or \(x > 1\).

Step 3: Solve the Second Inequality

Now solve the inequality \(2x - 5 < 7\).Add 5 to both sides:\[ 2x - 5 + 5 < 7 + 5 \]This simplifies to:\[ 2x < 12 \]Divide both sides by 2:\[ x < 6 \].

Step 4: Combine the Solutions

Combine the two solutions \(x > 1\) and \(x < 6\). The combined inequality is:\[ 1 < x < 6 \].This means the solution consists of all values for \(x\) that are greater than 1 and less than 6.

Step 5: Express the Solution as an Interval

Express the combined solution \(1 < x < 6\) as an interval.In interval notation, this is:\((1, 6)\).This interval represents all real numbers \(x\) that are greater than 1 and less than 6.

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities combined together. The two inequalities describe a range of values that satisfy both conditions simultaneously. For example, when you see \[-3 < 2x - 5 < 7\], you are dealing with a compound inequality. This expression means that we are looking for values of \(x\) that satisfy both \(-3 < 2x - 5\) and \(2x - 5 < 7\) at the same time.
Solving compound inequalities requires handling each inequality separately at first. Once you solve each one, you find the intersection of their solutions. This intersection gives you the range of values that make the entire inequality true. It is a crucial step because compound inequalities ensure our solution set fits all situations given by the individual inequalities.

Interval Notation

Interval notation is a method used to represent the set of solutions for an inequality succinctly. It includes values between two numbers and shows whether those boundary numbers are included in the set. With interval notation, we use parentheses \(( )\) to denote that an endpoint is not included and brackets \([ ]\) to denote that it is.
For the compound inequality \(1 < x < 6\), the solution is expressed as the interval \((1, 6)\). This means all numbers greater than 1 and less than 6 are solutions. If our inequality was \(1 \leq x < 6\), it would include 1, and the interval would be \([1, 6)\). Understanding interval notation is powerful for quickly communicating the solutions of inequalities.

Inequality Solutions

Solving inequalities involves finding the range of values for a variable that satisfy the given inequality. In the process of solving compound inequalities, like \[-3 < 2x - 5 < 7\], the goal is to isolate the variable by performing operations such as addition, subtraction, multiplication, or division. These operations are similar to those used in solving equations.

• Begin with isolating the term with the variable by adding or subtracting terms on both sides.
• Divide or multiply to completely solve for the variable.

There is a key difference when dealing with inequalities: multiplying or dividing by a negative number requires flipping the inequality sign. In our example, solving simultaneously revealed two conditions: \(x > 1\) and \(x < 6\). These conditions were combined seamlessly, resulting in the solution \(1 < x < 6\), representing values common to both expressions.