Question

type the correct answer in each box.
an industrial laser cuts steel plates that are 11.93 meters long. the laser can have an error value of at most 0.02 meters.
the inequality |x - | ≤ can be used to find the range of acceptable lengths for the steel plates.
the range of acceptable lengths for the steel plates is ≤ x ≤ .

Explanation:

Step1: Identify the target value

The target length of the steel plates is 11.93 meters. So in the absolute value inequality \(|x - a| \leq b\), \(a\) is the target value. So \(a = 11.93\).

Step2: Identify the error bound

The maximum error is 0.02 meters. So \(b = 0.02\). So the inequality is \(|x - 11.93| \leq 0.02\).

Step3: Solve the absolute value inequality

To solve \(|x - 11.93| \leq 0.02\), we use the property of absolute value inequalities: if \(|u| \leq c\) (where \(c\geq0\)), then \(-c \leq u \leq c\). Here \(u=x - 11.93\) and \(c = 0.02\). So we have \(- 0.02\leq x - 11.93\leq0.02\).

Step4: Find the lower bound

Add 11.93 to all parts of the inequality. For the left - hand side: \(x-11.93\geq - 0.02\) implies \(x\geq11.93 - 0.02=11.91\).

Step5: Find the upper bound

For the right - hand side: \(x - 11.93\leq0.02\) implies \(x\leq11.93 + 0.02 = 11.95\).

Answer:

The inequality \(|x - \boldsymbol{11.93}| \leq \boldsymbol{0.02}\) can be used to find the range of acceptable lengths for the steel plates. The range of acceptable lengths for the steel plates is \(\boldsymbol{11.91}\leq x\leq\boldsymbol{11.95}\).